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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
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@@ -371,7 +377,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -392,7 +398,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -469,7 +475,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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@@ -502,7 +508,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -604,7 +610,7 @@
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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@@ -637,7 +643,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
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<div data-md-component="skip">
<a href="#41-arrays" class="md-skip">
<a href="#41-array" class="md-skip">
Skip to content
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@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
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</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<label class="md-nav__title" for="__nav_5">
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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<li class="md-nav__item">
<a href="#411-common-operations-on-arrays" class="md-nav__link">
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4.1.1 &nbsp; Common Operations on Arrays
4.1.1 &nbsp; Common operations on arrays
</span>
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<nav class="md-nav" aria-label="4.1.1   Common operations on arrays">
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<a href="#1-initializing-arrays" class="md-nav__link">
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1. &nbsp; Initializing Arrays
1. &nbsp; Initializing arrays
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2. &nbsp; Accessing Elements
2. &nbsp; Accessing elements
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3. &nbsp; Inserting Elements
3. &nbsp; Inserting elements
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<a href="#4-deleting-elements" class="md-nav__link">
<span class="md-ellipsis">
4. &nbsp; Deleting Elements
4. &nbsp; Deleting elements
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<li class="md-nav__item">
<a href="#5-traversing-arrays" class="md-nav__link">
<span class="md-ellipsis">
5. &nbsp; Traversing Arrays
5. &nbsp; Traversing arrays
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<a href="#6-finding-elements" class="md-nav__link">
<span class="md-ellipsis">
6. &nbsp; Finding Elements
6. &nbsp; Finding elements
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<li class="md-nav__item">
<a href="#7-expanding-arrays" class="md-nav__link">
<span class="md-ellipsis">
7. &nbsp; Expanding Arrays
7. &nbsp; Expanding arrays
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<li class="md-nav__item">
<a href="#412-advantages-and-limitations-of-arrays" class="md-nav__link">
<span class="md-ellipsis">
4.1.2 &nbsp; Advantages and Limitations of Arrays
4.1.2 &nbsp; Advantages and limitations of arrays
</span>
</a>
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<li class="md-nav__item">
<a href="#413-typical-applications-of-arrays" class="md-nav__link">
<span class="md-ellipsis">
4.1.3 &nbsp; Typical Applications of Arrays
4.1.3 &nbsp; Typical applications of arrays
</span>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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5.3 Double-ended Queue
5.3 Double-ended queue
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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<li class="md-nav__item">
<a href="#411-common-operations-on-arrays" class="md-nav__link">
<span class="md-ellipsis">
4.1.1 &nbsp; Common Operations on Arrays
4.1.1 &nbsp; Common operations on arrays
</span>
</a>
<nav class="md-nav" aria-label="4.1.1   Common Operations on Arrays">
<nav class="md-nav" aria-label="4.1.1   Common operations on arrays">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#1-initializing-arrays" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Initializing Arrays
1. &nbsp; Initializing arrays
</span>
</a>
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<li class="md-nav__item">
<a href="#2-accessing-elements" class="md-nav__link">
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2. &nbsp; Accessing Elements
2. &nbsp; Accessing elements
</span>
</a>
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<a href="#3-inserting-elements" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Inserting Elements
3. &nbsp; Inserting elements
</span>
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<li class="md-nav__item">
<a href="#4-deleting-elements" class="md-nav__link">
<span class="md-ellipsis">
4. &nbsp; Deleting Elements
4. &nbsp; Deleting elements
</span>
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<a href="#5-traversing-arrays" class="md-nav__link">
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5. &nbsp; Traversing Arrays
5. &nbsp; Traversing arrays
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<a href="#6-finding-elements" class="md-nav__link">
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6. &nbsp; Finding Elements
6. &nbsp; Finding elements
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<a href="#7-expanding-arrays" class="md-nav__link">
<span class="md-ellipsis">
7. &nbsp; Expanding Arrays
7. &nbsp; Expanding arrays
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<a href="#412-advantages-and-limitations-of-arrays" class="md-nav__link">
<span class="md-ellipsis">
4.1.2 &nbsp; Advantages and Limitations of Arrays
4.1.2 &nbsp; Advantages and limitations of arrays
</span>
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<a href="#413-typical-applications-of-arrays" class="md-nav__link">
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4.1.3 &nbsp; Typical Applications of Arrays
4.1.3 &nbsp; Typical applications of arrays
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<!-- Page content -->
<h1 id="41-arrays">4.1 &nbsp; Arrays<a class="headerlink" href="#41-arrays" title="Permanent link">&para;</a></h1>
<h1 id="41-array">4.1 &nbsp; Array<a class="headerlink" href="#41-array" title="Permanent link">&para;</a></h1>
<p>An "array" is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an "index". Please refer to the Figure 4-1 to observe how arrays work and grasp these key terms.</p>
<p><a class="glightbox" href="../array.assets/array_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array Definition and Storage Method" class="animation-figure" src="../array.assets/array_definition.png" /></a></p>
<p align="center"> Figure 4-1 &nbsp; Array Definition and Storage Method </p>
<p><a class="glightbox" href="../array.assets/array_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array definition and storage method" class="animation-figure" src="../array.assets/array_definition.png" /></a></p>
<p align="center"> Figure 4-1 &nbsp; Array definition and storage method </p>
<h2 id="411-common-operations-on-arrays">4.1.1 &nbsp; Common Operations on Arrays<a class="headerlink" href="#411-common-operations-on-arrays" title="Permanent link">&para;</a></h2>
<h3 id="1-initializing-arrays">1. &nbsp; Initializing Arrays<a class="headerlink" href="#1-initializing-arrays" title="Permanent link">&para;</a></h3>
<h2 id="411-common-operations-on-arrays">4.1.1 &nbsp; Common operations on arrays<a class="headerlink" href="#411-common-operations-on-arrays" title="Permanent link">&para;</a></h2>
<h3 id="1-initializing-arrays">1. &nbsp; Initializing arrays<a class="headerlink" href="#1-initializing-arrays" title="Permanent link">&para;</a></h3>
<p>Arrays can be initialized in two ways depending on the needs: either without initial values or with specified initial values. When initial values are not specified, most programming languages will set the array elements to <span class="arithmatex">\(0\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Zig</label></div>
<div class="tabbed-content">
@@ -2321,10 +2327,10 @@
</div>
</div>
</div>
<h3 id="2-accessing-elements">2. &nbsp; Accessing Elements<a class="headerlink" href="#2-accessing-elements" title="Permanent link">&para;</a></h3>
<h3 id="2-accessing-elements">2. &nbsp; Accessing elements<a class="headerlink" href="#2-accessing-elements" title="Permanent link">&para;</a></h3>
<p>Elements in an array are stored in contiguous memory spaces, making it simpler to compute each element's memory address. The formula shown in the Figure below aids in determining an element's memory address, utilizing the array's memory address (specifically, the first element's address) and the element's index. This computation streamlines direct access to the desired element.</p>
<p><a class="glightbox" href="../array.assets/array_memory_location_calculation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory Address Calculation for Array Elements" class="animation-figure" src="../array.assets/array_memory_location_calculation.png" /></a></p>
<p align="center"> Figure 4-2 &nbsp; Memory Address Calculation for Array Elements </p>
<p><a class="glightbox" href="../array.assets/array_memory_location_calculation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory address calculation for array elements" class="animation-figure" src="../array.assets/array_memory_location_calculation.png" /></a></p>
<p align="center"> Figure 4-2 &nbsp; Memory address calculation for array elements </p>
<p>As observed in the above illustration, array indexing conventionally begins at <span class="arithmatex">\(0\)</span>. While this might appear counterintuitive, considering counting usually starts at <span class="arithmatex">\(1\)</span>, within the address calculation formula, <strong>an index is essentially an offset from the memory address</strong>. For the first element's address, this offset is <span class="arithmatex">\(0\)</span>, validating its index as <span class="arithmatex">\(0\)</span>.</p>
<p>Accessing elements in an array is highly efficient, allowing us to randomly access any element in <span class="arithmatex">\(O(1)\)</span> time.</p>
@@ -2491,10 +2497,10 @@
<p><div style="height: 531px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=import%20random%0A%0Adef%20random_access%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%9A%8F%E6%9C%BA%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%22%22%22%0A%20%20%20%20%23%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5B0,%20len%28nums%29-1%5D%20%E4%B8%AD%E9%9A%8F%E6%9C%BA%E6%8A%BD%E5%8F%96%E4%B8%80%E4%B8%AA%E6%95%B0%E5%AD%97%0A%20%20%20%20random_index%20%3D%20random.randint%280,%20len%28nums%29%20-%201%29%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%B9%B6%E8%BF%94%E5%9B%9E%E9%9A%8F%E6%9C%BA%E5%85%83%E7%B4%A0%0A%20%20%20%20random_num%20%3D%20nums%5Brandom_index%5D%0A%20%20%20%20return%20random_num%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E9%9A%8F%E6%9C%BA%E8%AE%BF%E9%97%AE%0A%20%20%20%20random_num%3A%20int%20%3D%20random_access%28nums%29%0A%20%20%20%20print%28%22%E5%9C%A8%20nums%20%E4%B8%AD%E8%8E%B7%E5%8F%96%E9%9A%8F%E6%9C%BA%E5%85%83%E7%B4%A0%22,%20random_num%29%0A&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20random%0A%0Adef%20random_access%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%9A%8F%E6%9C%BA%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%22%22%22%0A%20%20%20%20%23%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5B0,%20len%28nums%29-1%5D%20%E4%B8%AD%E9%9A%8F%E6%9C%BA%E6%8A%BD%E5%8F%96%E4%B8%80%E4%B8%AA%E6%95%B0%E5%AD%97%0A%20%20%20%20random_index%20%3D%20random.randint%280,%20len%28nums%29%20-%201%29%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%B9%B6%E8%BF%94%E5%9B%9E%E9%9A%8F%E6%9C%BA%E5%85%83%E7%B4%A0%0A%20%20%20%20random_num%20%3D%20nums%5Brandom_index%5D%0A%20%20%20%20return%20random_num%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E9%9A%8F%E6%9C%BA%E8%AE%BF%E9%97%AE%0A%20%20%20%20random_num%3A%20int%20%3D%20random_access%28nums%29%0A%20%20%20%20print%28%22%E5%9C%A8%20nums%20%E4%B8%AD%E8%8E%B7%E5%8F%96%E9%9A%8F%E6%9C%BA%E5%85%83%E7%B4%A0%22,%20random_num%29%0A&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h3 id="3-inserting-elements">3. &nbsp; Inserting Elements<a class="headerlink" href="#3-inserting-elements" title="Permanent link">&para;</a></h3>
<h3 id="3-inserting-elements">3. &nbsp; Inserting elements<a class="headerlink" href="#3-inserting-elements" title="Permanent link">&para;</a></h3>
<p>Array elements are tightly packed in memory, with no space available to accommodate additional data between them. Illustrated in Figure below, inserting an element in the middle of an array requires shifting all subsequent elements back by one position to create room for the new element.</p>
<p><a class="glightbox" href="../array.assets/array_insert_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array Element Insertion Example" class="animation-figure" src="../array.assets/array_insert_element.png" /></a></p>
<p align="center"> Figure 4-3 &nbsp; Array Element Insertion Example </p>
<p><a class="glightbox" href="../array.assets/array_insert_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array element insertion example" class="animation-figure" src="../array.assets/array_insert_element.png" /></a></p>
<p align="center"> Figure 4-3 &nbsp; Array element insertion example </p>
<p>It's important to note that due to the fixed length of an array, inserting an element will unavoidably result in the loss of the last element in the array. Solutions to address this issue will be explored in the "List" chapter.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:14"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><input id="__tabbed_3_14" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label><label for="__tabbed_3_14">Zig</label></div>
@@ -2674,10 +2680,10 @@
<p><div style="height: 495px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20insert%28nums%3A%20list%5Bint%5D,%20num%3A%20int,%20index%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E6%95%B0%E7%BB%84%E7%9A%84%E7%B4%A2%E5%BC%95%20index%20%E5%A4%84%E6%8F%92%E5%85%A5%E5%85%83%E7%B4%A0%20num%22%22%22%0A%20%20%20%20%23%20%E6%8A%8A%E7%B4%A2%E5%BC%95%20index%20%E4%BB%A5%E5%8F%8A%E4%B9%8B%E5%90%8E%E7%9A%84%E6%89%80%E6%9C%89%E5%85%83%E7%B4%A0%E5%90%91%E5%90%8E%E7%A7%BB%E5%8A%A8%E4%B8%80%E4%BD%8D%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%20index,%20-1%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20nums%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E5%B0%86%20num%20%E8%B5%8B%E7%BB%99%20index%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0%0A%20%20%20%20nums%5Bindex%5D%20%3D%20num%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E5%85%83%E7%B4%A0%0A%20%20%20%20insert%28nums,%206,%203%29%0A%20%20%20%20print%28%22%E5%9C%A8%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E6%8F%92%E5%85%A5%E6%95%B0%E5%AD%97%206%20%EF%BC%8C%E5%BE%97%E5%88%B0%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20insert%28nums%3A%20list%5Bint%5D,%20num%3A%20int,%20index%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E6%95%B0%E7%BB%84%E7%9A%84%E7%B4%A2%E5%BC%95%20index%20%E5%A4%84%E6%8F%92%E5%85%A5%E5%85%83%E7%B4%A0%20num%22%22%22%0A%20%20%20%20%23%20%E6%8A%8A%E7%B4%A2%E5%BC%95%20index%20%E4%BB%A5%E5%8F%8A%E4%B9%8B%E5%90%8E%E7%9A%84%E6%89%80%E6%9C%89%E5%85%83%E7%B4%A0%E5%90%91%E5%90%8E%E7%A7%BB%E5%8A%A8%E4%B8%80%E4%BD%8D%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%20index,%20-1%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20nums%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E5%B0%86%20num%20%E8%B5%8B%E7%BB%99%20index%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0%0A%20%20%20%20nums%5Bindex%5D%20%3D%20num%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E5%85%83%E7%B4%A0%0A%20%20%20%20insert%28nums,%206,%203%29%0A%20%20%20%20print%28%22%E5%9C%A8%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E6%8F%92%E5%85%A5%E6%95%B0%E5%AD%97%206%20%EF%BC%8C%E5%BE%97%E5%88%B0%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h3 id="4-deleting-elements">4. &nbsp; Deleting Elements<a class="headerlink" href="#4-deleting-elements" title="Permanent link">&para;</a></h3>
<h3 id="4-deleting-elements">4. &nbsp; Deleting elements<a class="headerlink" href="#4-deleting-elements" title="Permanent link">&para;</a></h3>
<p>Similarly, as depicted in the Figure 4-4 , to delete an element at index <span class="arithmatex">\(i\)</span>, all elements following index <span class="arithmatex">\(i\)</span> must be moved forward by one position.</p>
<p><a class="glightbox" href="../array.assets/array_remove_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array Element Deletion Example" class="animation-figure" src="../array.assets/array_remove_element.png" /></a></p>
<p align="center"> Figure 4-4 &nbsp; Array Element Deletion Example </p>
<p><a class="glightbox" href="../array.assets/array_remove_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array element deletion example" class="animation-figure" src="../array.assets/array_remove_element.png" /></a></p>
<p align="center"> Figure 4-4 &nbsp; Array element deletion example </p>
<p>Please note that after deletion, the former last element becomes "meaningless," hence requiring no specific modification.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label><label for="__tabbed_4_14">Zig</label></div>
@@ -2831,11 +2837,11 @@
</details>
<p>In summary, the insertion and deletion operations in arrays present the following disadvantages:</p>
<ul>
<li><strong>High Time Complexity</strong>: Both insertion and deletion in an array have an average time complexity of <span class="arithmatex">\(O(n)\)</span>, where <span class="arithmatex">\(n\)</span> is the length of the array.</li>
<li><strong>Loss of Elements</strong>: Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.</li>
<li><strong>Waste of Memory</strong>: Initializing a longer array and utilizing only the front part results in "meaningless" end elements during insertion, leading to some wasted memory space.</li>
<li><strong>High time complexity</strong>: Both insertion and deletion in an array have an average time complexity of <span class="arithmatex">\(O(n)\)</span>, where <span class="arithmatex">\(n\)</span> is the length of the array.</li>
<li><strong>Loss of elements</strong>: Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.</li>
<li><strong>Waste of memory</strong>: Initializing a longer array and utilizing only the front part results in "meaningless" end elements during insertion, leading to some wasted memory space.</li>
</ul>
<h3 id="5-traversing-arrays">5. &nbsp; Traversing Arrays<a class="headerlink" href="#5-traversing-arrays" title="Permanent link">&para;</a></h3>
<h3 id="5-traversing-arrays">5. &nbsp; Traversing arrays<a class="headerlink" href="#5-traversing-arrays" title="Permanent link">&para;</a></h3>
<p>In most programming languages, we can traverse an array either by using indices or by directly iterating over each element:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:14"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label><label for="__tabbed_5_14">Zig</label></div>
<div class="tabbed-content">
@@ -3068,7 +3074,7 @@
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20traverse%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22%E9%81%8D%E5%8E%86%E6%95%B0%E7%BB%84%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20%E9%80%9A%E8%BF%87%E7%B4%A2%E5%BC%95%E9%81%8D%E5%8E%86%E6%95%B0%E7%BB%84%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%20%20%20%20%23%20%E7%9B%B4%E6%8E%A5%E9%81%8D%E5%8E%86%E6%95%B0%E7%BB%84%E5%85%83%E7%B4%A0%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num%0A%20%20%20%20%23%20%E5%90%8C%E6%97%B6%E9%81%8D%E5%8E%86%E6%95%B0%E6%8D%AE%E7%B4%A2%E5%BC%95%E5%92%8C%E5%85%83%E7%B4%A0%0A%20%20%20%20for%20i,%20num%20in%20enumerate%28nums%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E6%95%B0%E7%BB%84%0A%20%20%20%20traverse%28nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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</details>
<h3 id="6-finding-elements">6. &nbsp; Finding Elements<a class="headerlink" href="#6-finding-elements" title="Permanent link">&para;</a></h3>
<h3 id="6-finding-elements">6. &nbsp; Finding elements<a class="headerlink" href="#6-finding-elements" title="Permanent link">&para;</a></h3>
<p>Locating a specific element within an array involves iterating through the array, checking each element to determine if it matches the desired value.</p>
<p>Because arrays are linear data structures, this operation is commonly referred to as "linear search."</p>
<div class="tabbed-set tabbed-alternate" data-tabs="6:14"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><input id="__tabbed_6_14" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label><label for="__tabbed_6_14">Zig</label></div>
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<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20find%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E6%95%B0%E7%BB%84%E4%B8%AD%E6%9F%A5%E6%89%BE%E6%8C%87%E5%AE%9A%E5%85%83%E7%B4%A0%22%22%22%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E6%9F%A5%E6%89%BE%E5%85%83%E7%B4%A0%0A%20%20%20%20index%3A%20int%20%3D%20find%28nums,%203%29%0A%20%20%20%20print%28%22%E5%9C%A8%20nums%20%E4%B8%AD%E6%9F%A5%E6%89%BE%E5%85%83%E7%B4%A0%203%20%EF%BC%8C%E5%BE%97%E5%88%B0%E7%B4%A2%E5%BC%95%20%3D%22,%20index%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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</details>
<h3 id="7-expanding-arrays">7. &nbsp; Expanding Arrays<a class="headerlink" href="#7-expanding-arrays" title="Permanent link">&para;</a></h3>
<h3 id="7-expanding-arrays">7. &nbsp; Expanding arrays<a class="headerlink" href="#7-expanding-arrays" title="Permanent link">&para;</a></h3>
<p>In complex system environments, ensuring the availability of memory space after an array for safe capacity extension becomes challenging. Consequently, in most programming languages, <strong>the length of an array is immutable</strong>.</p>
<p>To expand an array, it's necessary to create a larger array and then copy the elements from the original array. This operation has a time complexity of <span class="arithmatex">\(O(n)\)</span> and can be time-consuming for large arrays. The code are as follows:</p>
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<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=%23%20%E8%AF%B7%E6%B3%A8%E6%84%8F%EF%BC%8CPython%20%E7%9A%84%20list%20%E6%98%AF%E5%8A%A8%E6%80%81%E6%95%B0%E7%BB%84%EF%BC%8C%E5%8F%AF%E4%BB%A5%E7%9B%B4%E6%8E%A5%E6%89%A9%E5%B1%95%0A%23%20%E4%B8%BA%E4%BA%86%E6%96%B9%E4%BE%BF%E5%AD%A6%E4%B9%A0%EF%BC%8C%E6%9C%AC%E5%87%BD%E6%95%B0%E5%B0%86%20list%20%E7%9C%8B%E4%BD%9C%E9%95%BF%E5%BA%A6%E4%B8%8D%E5%8F%AF%E5%8F%98%E7%9A%84%E6%95%B0%E7%BB%84%0Adef%20extend%28nums%3A%20list%5Bint%5D,%20enlarge%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22%E6%89%A9%E5%B1%95%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%22%22%22%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%B8%80%E4%B8%AA%E6%89%A9%E5%B1%95%E9%95%BF%E5%BA%A6%E5%90%8E%E7%9A%84%E6%95%B0%E7%BB%84%0A%20%20%20%20res%20%3D%20%5B0%5D%20*%20%28len%28nums%29%20%2B%20enlarge%29%0A%20%20%20%20%23%20%E5%B0%86%E5%8E%9F%E6%95%B0%E7%BB%84%E4%B8%AD%E7%9A%84%E6%89%80%E6%9C%89%E5%85%83%E7%B4%A0%E5%A4%8D%E5%88%B6%E5%88%B0%E6%96%B0%E6%95%B0%E7%BB%84%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20res%5Bi%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E6%89%A9%E5%B1%95%E5%90%8E%E7%9A%84%E6%96%B0%E6%95%B0%E7%BB%84%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22%E6%95%B0%E7%BB%84%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20%E9%95%BF%E5%BA%A6%E6%89%A9%E5%B1%95%0A%20%20%20%20nums%20%3D%20extend%28nums,%203%29%0A%20%20%20%20print%28%22%E5%B0%86%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%E6%89%A9%E5%B1%95%E8%87%B3%208%20%EF%BC%8C%E5%BE%97%E5%88%B0%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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</details>
<h2 id="412-advantages-and-limitations-of-arrays">4.1.2 &nbsp; Advantages and Limitations of Arrays<a class="headerlink" href="#412-advantages-and-limitations-of-arrays" title="Permanent link">&para;</a></h2>
<h2 id="412-advantages-and-limitations-of-arrays">4.1.2 &nbsp; Advantages and limitations of arrays<a class="headerlink" href="#412-advantages-and-limitations-of-arrays" title="Permanent link">&para;</a></h2>
<p>Arrays are stored in contiguous memory spaces and consist of elements of the same type. This approach provides substantial prior information that systems can leverage to optimize the efficiency of data structure operations.</p>
<ul>
<li><strong>High Space Efficiency</strong>: Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.</li>
<li><strong>Support for Random Access</strong>: Arrays allow <span class="arithmatex">\(O(1)\)</span> time access to any element.</li>
<li><strong>Cache Locality</strong>: When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.</li>
<li><strong>High space efficiency</strong>: Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.</li>
<li><strong>Support for random access</strong>: Arrays allow <span class="arithmatex">\(O(1)\)</span> time access to any element.</li>
<li><strong>Cache locality</strong>: When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.</li>
</ul>
<p>However, continuous space storage is a double-edged sword, with the following limitations:</p>
<ul>
<li><strong>Low Efficiency in Insertion and Deletion</strong>: As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.</li>
<li><strong>Fixed Length</strong>: The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.</li>
<li><strong>Space Wastage</strong>: If the allocated array size exceeds the what is necessary, the extra space is wasted.</li>
<li><strong>Low efficiency in insertion and deletion</strong>: As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.</li>
<li><strong>Fixed length</strong>: The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.</li>
<li><strong>Space wastage</strong>: If the allocated array size exceeds the what is necessary, the extra space is wasted.</li>
</ul>
<h2 id="413-typical-applications-of-arrays">4.1.3 &nbsp; Typical Applications of Arrays<a class="headerlink" href="#413-typical-applications-of-arrays" title="Permanent link">&para;</a></h2>
<h2 id="413-typical-applications-of-arrays">4.1.3 &nbsp; Typical applications of arrays<a class="headerlink" href="#413-typical-applications-of-arrays" title="Permanent link">&para;</a></h2>
<p>Arrays are fundamental and widely used data structures. They find frequent application in various algorithms and serve in the implementation of complex data structures.</p>
<ul>
<li><strong>Random Access</strong>: Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.</li>
<li><strong>Sorting and Searching</strong>: Arrays are the most commonly used data structure for sorting and searching algorithms. Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.</li>
<li><strong>Lookup Tables</strong>: Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.</li>
<li><strong>Machine Learning</strong>: Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.</li>
<li><strong>Data Structure Implementation</strong>: Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li>
<li><strong>Random access</strong>: Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.</li>
<li><strong>Sorting and searching</strong>: Arrays are the most commonly used data structure for sorting and searching algorithms. Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.</li>
<li><strong>Lookup tables</strong>: Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.</li>
<li><strong>Machine learning</strong>: Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.</li>
<li><strong>Data structure implementation</strong>: Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li>
</ul>
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Chapter 4. &nbsp; Arrays and Linked Lists
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</span>
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
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<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -982,7 +988,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -998,7 +1004,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1036,7 +1042,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1078,7 +1084,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1157,7 +1163,7 @@
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1173,7 +1179,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1232,7 +1238,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1311,7 +1317,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1327,7 +1333,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1344,7 +1350,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1365,7 +1371,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1386,7 +1392,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2003,8 +2009,8 @@
<!-- Page content -->
<h1 id="chapter-4-arrays-and-linked-lists">Chapter 4. &nbsp; Arrays and Linked Lists<a class="headerlink" href="#chapter-4-arrays-and-linked-lists" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_array_and_linkedlist.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Arrays and Linked Lists" class="cover-image" src="../assets/covers/chapter_array_and_linkedlist.jpg" /></a></p>
<h1 id="chapter-4-arrays-and-linked-lists">Chapter 4. &nbsp; Arrays and linked lists<a class="headerlink" href="#chapter-4-arrays-and-linked-lists" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_array_and_linkedlist.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Arrays and linked lists" class="cover-image" src="../assets/covers/chapter_array_and_linkedlist.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>The world of data structures resembles a sturdy brick wall.</p>
@@ -2013,9 +2019,9 @@
<h2 id="chapter-contents">Chapter Contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/array/">4.1 &nbsp; Array</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/linked_list/">4.2 &nbsp; Linked List</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/linked_list/">4.2 &nbsp; Linked list</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/list/">4.3 &nbsp; List</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/ram_and_cache/">4.4 &nbsp; Memory and Cache</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/ram_and_cache/">4.4 &nbsp; Memory and cache</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_array_and_linkedlist/summary/">4.5 &nbsp; Summary</a></li>
</ul>
en/chapter_array_and_linkedlist/linked_list/index.html
+107 -101
View File
@@ -26,7 +26,7 @@
<title>4.2 Linked List - Hello Algo</title>
<title>4.2 Linked list - Hello Algo</title>
@@ -91,7 +91,7 @@
<div data-md-component="skip">
<a href="#42-linked-lists" class="md-skip">
<a href="#42-linked-list" class="md-skip">
Skip to content
</a>
@@ -153,7 +153,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
</div>
@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -982,7 +988,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -998,7 +1004,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1045,7 +1051,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1056,7 +1062,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1080,35 +1086,35 @@
<li class="md-nav__item">
<a href="#421-common-operations-on-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.1 &nbsp; Common Operations on Linked Lists
4.2.1 &nbsp; Common operations on linked lists
</span>
</a>
<nav class="md-nav" aria-label="4.2.1   Common Operations on Linked Lists">
<nav class="md-nav" aria-label="4.2.1   Common operations on linked lists">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#1-initializing-a-linked-list" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Initializing a Linked List
1. &nbsp; Initializing a linked list
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#2-inserting-a-node" class="md-nav__link">
<a href="#2-inserting-nodes" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Inserting a Node
2. &nbsp; Inserting nodes
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#3-deleting-a-node" class="md-nav__link">
<a href="#3-deleting-nodes" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Deleting a Node
3. &nbsp; Deleting nodes
</span>
</a>
@@ -1117,7 +1123,7 @@
<li class="md-nav__item">
<a href="#4-accessing-nodes" class="md-nav__link">
<span class="md-ellipsis">
4. &nbsp; Accessing Nodes
4. &nbsp; Accessing nodes
</span>
</a>
@@ -1126,7 +1132,7 @@
<li class="md-nav__item">
<a href="#5-finding-nodes" class="md-nav__link">
<span class="md-ellipsis">
5. &nbsp; Finding Nodes
5. &nbsp; Finding nodes
</span>
</a>
@@ -1140,7 +1146,7 @@
<li class="md-nav__item">
<a href="#422-arrays-vs-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.2 &nbsp; Arrays vs. Linked Lists
4.2.2 &nbsp; Arrays vs. linked lists
</span>
</a>
@@ -1149,7 +1155,7 @@
<li class="md-nav__item">
<a href="#423-common-types-of-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.3 &nbsp; Common Types of Linked Lists
4.2.3 &nbsp; Common types of linked lists
</span>
</a>
@@ -1158,7 +1164,7 @@
<li class="md-nav__item">
<a href="#424-typical-applications-of-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.4 &nbsp; Typical Applications of Linked Lists
4.2.4 &nbsp; Typical applications of linked lists
</span>
</a>
@@ -1205,7 +1211,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1284,7 +1290,7 @@
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1300,7 +1306,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1359,7 +1365,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1438,7 +1444,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1454,7 +1460,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1471,7 +1477,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1492,7 +1498,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1513,7 +1519,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2088,35 +2094,35 @@
<li class="md-nav__item">
<a href="#421-common-operations-on-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.1 &nbsp; Common Operations on Linked Lists
4.2.1 &nbsp; Common operations on linked lists
</span>
</a>
<nav class="md-nav" aria-label="4.2.1   Common Operations on Linked Lists">
<nav class="md-nav" aria-label="4.2.1   Common operations on linked lists">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#1-initializing-a-linked-list" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Initializing a Linked List
1. &nbsp; Initializing a linked list
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#2-inserting-a-node" class="md-nav__link">
<a href="#2-inserting-nodes" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Inserting a Node
2. &nbsp; Inserting nodes
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#3-deleting-a-node" class="md-nav__link">
<a href="#3-deleting-nodes" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Deleting a Node
3. &nbsp; Deleting nodes
</span>
</a>
@@ -2125,7 +2131,7 @@
<li class="md-nav__item">
<a href="#4-accessing-nodes" class="md-nav__link">
<span class="md-ellipsis">
4. &nbsp; Accessing Nodes
4. &nbsp; Accessing nodes
</span>
</a>
@@ -2134,7 +2140,7 @@
<li class="md-nav__item">
<a href="#5-finding-nodes" class="md-nav__link">
<span class="md-ellipsis">
5. &nbsp; Finding Nodes
5. &nbsp; Finding nodes
</span>
</a>
@@ -2148,7 +2154,7 @@
<li class="md-nav__item">
<a href="#422-arrays-vs-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.2 &nbsp; Arrays vs. Linked Lists
4.2.2 &nbsp; Arrays vs. linked lists
</span>
</a>
@@ -2157,7 +2163,7 @@
<li class="md-nav__item">
<a href="#423-common-types-of-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.3 &nbsp; Common Types of Linked Lists
4.2.3 &nbsp; Common types of linked lists
</span>
</a>
@@ -2166,7 +2172,7 @@
<li class="md-nav__item">
<a href="#424-typical-applications-of-linked-lists" class="md-nav__link">
<span class="md-ellipsis">
4.2.4 &nbsp; Typical Applications of Linked Lists
4.2.4 &nbsp; Typical applications of linked lists
</span>
</a>
@@ -2208,12 +2214,12 @@
<!-- Page content -->
<h1 id="42-linked-lists">4.2 &nbsp; Linked Lists<a class="headerlink" href="#42-linked-lists" title="Permanent link">&para;</a></h1>
<h1 id="42-linked-list">4.2 &nbsp; Linked list<a class="headerlink" href="#42-linked-list" title="Permanent link">&para;</a></h1>
<p>Memory space is a shared resource among all programs. In a complex system environment, available memory can be dispersed throughout the memory space. We understand that the memory allocated for an array must be continuous. However, for very large arrays, finding a sufficiently large contiguous memory space might be challenging. This is where the flexible advantage of linked lists becomes evident.</p>
<p>A "linked list" is a linear data structure in which each element is a node object, and the nodes are interconnected through "references". These references hold the memory addresses of subsequent nodes, enabling navigation from one node to the next.</p>
<p>The design of linked lists allows for their nodes to be distributed across memory locations without requiring contiguous memory addresses.</p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked List Definition and Storage Method" class="animation-figure" src="../linked_list.assets/linkedlist_definition.png" /></a></p>
<p align="center"> Figure 4-5 &nbsp; Linked List Definition and Storage Method </p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked list definition and storage method" class="animation-figure" src="../linked_list.assets/linkedlist_definition.png" /></a></p>
<p align="center"> Figure 4-5 &nbsp; Linked list definition and storage method </p>
<p>As shown in the figure, we see that the basic building block of a linked list is the "node" object. Each node comprises two key components: the node's "value" and a "reference" to the next node.</p>
<ul>
@@ -2226,14 +2232,14 @@
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span> <span class="nc">ListNode</span><span class="p">:</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Linked List Node Class&quot;&quot;&quot;</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Linked list node class&quot;&quot;&quot;</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span> <span class="c1"># Node value</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># Reference to the next node</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Linked List Node Structure */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Linked list node structure */</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the next node</span>
@@ -2242,7 +2248,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Linked List Node Class */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kd">class</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
@@ -2251,7 +2257,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Linked List Node Class */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Constructor</span>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
@@ -2259,7 +2265,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Linked List Node Structure */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Linked list node structure */</span>
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w"> </span><span class="nx">Val</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="c1">// Node value</span>
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="w"> </span><span class="c1">// Pointer to the next node</span>
@@ -2275,7 +2281,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Linked List Node Class */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">class</span> <span class="nc">ListNode</span> <span class="p">{</span>
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a> <span class="kd">var</span> <span class="nv">val</span><span class="p">:</span> <span class="nb">Int</span> <span class="c1">// Node value</span>
<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a> <span class="kd">var</span> <span class="nv">next</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">?</span> <span class="c1">// Reference to the next node</span>
@@ -2287,7 +2293,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Linked List Node Class */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w"> </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">next</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Node value</span>
@@ -2297,7 +2303,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Linked List Node Class */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span>
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="nx">next</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
@@ -2309,7 +2315,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 链表节点类 */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
@@ -2320,7 +2326,7 @@
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span>::<span class="n">rc</span>::<span class="n">Rc</span><span class="p">;</span>
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span>::<span class="n">cell</span>::<span class="n">RefCell</span><span class="p">;</span>
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="cm">/* Linked List Node Class */</span>
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="cm">/* Linked list node class */</span>
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="cp">#[derive(Debug)]</span>
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="k">struct</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w"> </span><span class="n">val</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// Node value</span>
@@ -2329,7 +2335,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Linked List Node Structure */</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Linked list node structure */</span>
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the next node</span>
@@ -2350,7 +2356,7 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">// Linked List Node Class</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">// Linked list node class</span>
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="kr">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="kr">comptime</span><span class="w"> </span><span class="n">T</span><span class="o">:</span><span class="w"> </span><span class="kt">type</span><span class="p">)</span><span class="w"> </span><span class="kt">type</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w"> </span><span class="kr">const</span><span class="w"> </span><span class="n">Self</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@This</span><span class="p">();</span>
@@ -2369,8 +2375,8 @@
</div>
</div>
</div>
<h2 id="421-common-operations-on-linked-lists">4.2.1 &nbsp; Common Operations on Linked Lists<a class="headerlink" href="#421-common-operations-on-linked-lists" title="Permanent link">&para;</a></h2>
<h3 id="1-initializing-a-linked-list">1. &nbsp; Initializing a Linked List<a class="headerlink" href="#1-initializing-a-linked-list" title="Permanent link">&para;</a></h3>
<h2 id="421-common-operations-on-linked-lists">4.2.1 &nbsp; Common operations on linked lists<a class="headerlink" href="#421-common-operations-on-linked-lists" title="Permanent link">&para;</a></h2>
<h3 id="1-initializing-a-linked-list">1. &nbsp; Initializing a linked list<a class="headerlink" href="#1-initializing-a-linked-list" title="Permanent link">&para;</a></h3>
<p>Constructing a linked list is a two-step process: first, initializing each node object, and second, forming the reference links between the nodes. After initialization, we can traverse all nodes sequentially from the head node by following the <code>next</code> reference.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Zig</label></div>
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</div>
</div>
<p>The array as a whole is a variable, for instance, the array <code>nums</code> includes elements like <code>nums[0]</code>, <code>nums[1]</code>, and so on, whereas a linked list is made up of several distinct node objects. <strong>We typically refer to a linked list by its head node</strong>, for example, the linked list in the previous code snippet is referred to as <code>n0</code>.</p>
<h3 id="2-inserting-a-node">2. &nbsp; Inserting a Node<a class="headerlink" href="#2-inserting-a-node" title="Permanent link">&para;</a></h3>
<h3 id="2-inserting-nodes">2. &nbsp; Inserting nodes<a class="headerlink" href="#2-inserting-nodes" title="Permanent link">&para;</a></h3>
<p>Inserting a node into a linked list is very easy. As shown in the figure, let's assume we aim to insert a new node <code>P</code> between two adjacent nodes <code>n0</code> and <code>n1</code>. <strong>This can be achieved by simply modifying two node references (pointers)</strong>, with a time complexity of <span class="arithmatex">\(O(1)\)</span>.</p>
<p>By comparison, inserting an element into an array has a time complexity of <span class="arithmatex">\(O(n)\)</span>, which becomes less efficient when dealing with large data volumes.</p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_insert_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked List Node Insertion Example" class="animation-figure" src="../linked_list.assets/linkedlist_insert_node.png" /></a></p>
<p align="center"> Figure 4-6 &nbsp; Linked List Node Insertion Example </p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_insert_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked list node insertion example" class="animation-figure" src="../linked_list.assets/linkedlist_insert_node.png" /></a></p>
<p align="center"> Figure 4-6 &nbsp; Linked list node insertion example </p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:14"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><input id="__tabbed_3_14" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label><label for="__tabbed_3_14">Zig</label></div>
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@@ -2704,11 +2710,11 @@
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20insert%28n0%3A%20ListNode,%20P%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%22%22%22%0A%20%20%20%20n1%20%3D%20n0.next%0A%20%20%20%20P.next%20%3D%20n1%0A%20%20%20%20n0.next%20%3D%20P%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20ListNode%280%29%0A%20%20%20%20insert%28n0,%20p%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20insert%28n0%3A%20ListNode,%20P%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%22%22%22%0A%20%20%20%20n1%20%3D%20n0.next%0A%20%20%20%20P.next%20%3D%20n1%0A%20%20%20%20n0.next%20%3D%20P%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20ListNode%280%29%0A%20%20%20%20insert%28n0,%20p%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h3 id="3-deleting-a-node">3. &nbsp; Deleting a Node<a class="headerlink" href="#3-deleting-a-node" title="Permanent link">&para;</a></h3>
<h3 id="3-deleting-nodes">3. &nbsp; Deleting nodes<a class="headerlink" href="#3-deleting-nodes" title="Permanent link">&para;</a></h3>
<p>As shown in the figure, deleting a node from a linked list is also very easy, <strong>involving only the modification of a single node's reference (pointer)</strong>.</p>
<p>It's important to note that even though node <code>P</code> continues to point to <code>n1</code> after being deleted, it becomes inaccessible during linked list traversal. This effectively means that <code>P</code> is no longer a part of the linked list.</p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_remove_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked List Node Deletion" class="animation-figure" src="../linked_list.assets/linkedlist_remove_node.png" /></a></p>
<p align="center"> Figure 4-7 &nbsp; Linked List Node Deletion </p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_remove_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked list node deletion" class="animation-figure" src="../linked_list.assets/linkedlist_remove_node.png" /></a></p>
<p align="center"> Figure 4-7 &nbsp; Linked list node deletion </p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label><label for="__tabbed_4_14">Zig</label></div>
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@@ -2892,7 +2898,7 @@
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20remove%28n0%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%88%A0%E9%99%A4%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E7%9A%84%E9%A6%96%E4%B8%AA%E8%8A%82%E7%82%B9%22%22%22%0A%20%20%20%20if%20not%20n0.next%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20n0%20-%3E%20P%20-%3E%20n1%0A%20%20%20%20P%20%3D%20n0.next%0A%20%20%20%20n1%20%3D%20P.next%0A%20%20%20%20n0.next%20%3D%20n1%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20remove%28n0%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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</details>
<h3 id="4-accessing-nodes">4. &nbsp; Accessing Nodes<a class="headerlink" href="#4-accessing-nodes" title="Permanent link">&para;</a></h3>
<h3 id="4-accessing-nodes">4. &nbsp; Accessing nodes<a class="headerlink" href="#4-accessing-nodes" title="Permanent link">&para;</a></h3>
<p><strong>Accessing nodes in a linked list is less efficient</strong>. As previously mentioned, any element in an array can be accessed in <span class="arithmatex">\(O(1)\)</span> time. In contrast, with a linked list, the program involves starting from the head node and sequentially traversing through the nodes until the desired node is found. In other words, to access the <span class="arithmatex">\(i\)</span>-th node in a linked list, the program must iterate through <span class="arithmatex">\(i - 1\)</span> nodes, resulting in a time complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
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<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20access%28head%3A%20ListNode,%20index%3A%20int%29%20-%3E%20ListNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E8%AE%BF%E9%97%AE%E9%93%BE%E8%A1%A8%E4%B8%AD%E7%B4%A2%E5%BC%95%E4%B8%BA%20index%20%E7%9A%84%E8%8A%82%E7%82%B9%22%22%22%0A%20%20%20%20for%20_%20in%20range%28index%29%3A%0A%20%20%20%20%20%20%20%20if%20not%20head%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20return%20head%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E8%8A%82%E7%82%B9%0A%20%20%20%20node%20%3D%20access%28n0,%203%29%0A%20%20%20%20print%28%22%E9%93%BE%E8%A1%A8%E4%B8%AD%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E7%9A%84%E8%8A%82%E7%82%B9%E7%9A%84%E5%80%BC%20%3D%20%7B%7D%22.format%28node.val%29%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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</details>
<h3 id="5-finding-nodes">5. &nbsp; Finding Nodes<a class="headerlink" href="#5-finding-nodes" title="Permanent link">&para;</a></h3>
<h3 id="5-finding-nodes">5. &nbsp; Finding nodes<a class="headerlink" href="#5-finding-nodes" title="Permanent link">&para;</a></h3>
<p>Traverse the linked list to locate a node whose value matches <code>target</code>, and then output the index of that node within the linked list. This procedure is also an example of linear search. The corresponding code is provided below:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="6:14"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><input id="__tabbed_6_14" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label><label for="__tabbed_6_14">Zig</label></div>
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@@ -3285,9 +3291,9 @@
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20find%28head%3A%20ListNode,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E9%93%BE%E8%A1%A8%E4%B8%AD%E6%9F%A5%E6%89%BE%E5%80%BC%E4%B8%BA%20target%20%E7%9A%84%E9%A6%96%E4%B8%AA%E8%8A%82%E7%82%B9%22%22%22%0A%20%20%20%20index%20%3D%200%0A%20%20%20%20while%20head%3A%0A%20%20%20%20%20%20%20%20if%20head.val%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20index%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20%20%20%20%20index%20%2B%3D%201%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E6%9F%A5%E6%89%BE%E8%8A%82%E7%82%B9%0A%20%20%20%20index%20%3D%20find%28n0,%202%29%0A%20%20%20%20print%28%22%E9%93%BE%E8%A1%A8%E4%B8%AD%E5%80%BC%E4%B8%BA%202%20%E7%9A%84%E8%8A%82%E7%82%B9%E7%9A%84%E7%B4%A2%E5%BC%95%20%3D%20%7B%7D%22.format%28index%29%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20find%28head%3A%20ListNode,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E9%93%BE%E8%A1%A8%E4%B8%AD%E6%9F%A5%E6%89%BE%E5%80%BC%E4%B8%BA%20target%20%E7%9A%84%E9%A6%96%E4%B8%AA%E8%8A%82%E7%82%B9%22%22%22%0A%20%20%20%20index%20%3D%200%0A%20%20%20%20while%20head%3A%0A%20%20%20%20%20%20%20%20if%20head.val%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20index%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20%20%20%20%20index%20%2B%3D%201%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E6%9F%A5%E6%89%BE%E8%8A%82%E7%82%B9%0A%20%20%20%20index%20%3D%20find%28n0,%202%29%0A%20%20%20%20print%28%22%E9%93%BE%E8%A1%A8%E4%B8%AD%E5%80%BC%E4%B8%BA%202%20%E7%9A%84%E8%8A%82%E7%82%B9%E7%9A%84%E7%B4%A2%E5%BC%95%20%3D%20%7B%7D%22.format%28index%29%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h2 id="422-arrays-vs-linked-lists">4.2.2 &nbsp; Arrays vs. Linked Lists<a class="headerlink" href="#422-arrays-vs-linked-lists" title="Permanent link">&para;</a></h2>
<h2 id="422-arrays-vs-linked-lists">4.2.2 &nbsp; Arrays vs. linked lists<a class="headerlink" href="#422-arrays-vs-linked-lists" title="Permanent link">&para;</a></h2>
<p>The Table 4-1 summarizes the characteristics of arrays and linked lists, and it also compares their efficiencies in various operations. Because they utilize opposing storage strategies, their respective properties and operational efficiencies exhibit distinct contrasts.</p>
<p align="center"> Table 4-1 &nbsp; Efficiency Comparison of Arrays and Linked Lists </p>
<p align="center"> Table 4-1 &nbsp; Efficiency comparison of arrays and linked lists </p>
<div class="center-table">
<table>
@@ -3332,18 +3338,18 @@
</tbody>
</table>
</div>
<h2 id="423-common-types-of-linked-lists">4.2.3 &nbsp; Common Types of Linked Lists<a class="headerlink" href="#423-common-types-of-linked-lists" title="Permanent link">&para;</a></h2>
<h2 id="423-common-types-of-linked-lists">4.2.3 &nbsp; Common types of linked lists<a class="headerlink" href="#423-common-types-of-linked-lists" title="Permanent link">&para;</a></h2>
<p>As shown in the figure, there are three common types of linked lists.</p>
<ul>
<li><strong>Singly Linked List</strong>: This is the standard linked list described earlier. Nodes in a singly linked list include a value and a reference to the next node. The first node is known as the head node, and the last node, which points to null (<code>None</code>), is the tail node.</li>
<li><strong>Circular Linked List</strong>: This is formed when the tail node of a singly linked list points back to the head node, creating a loop. In a circular linked list, any node can function as the head node.</li>
<li><strong>Doubly Linked List</strong>: In contrast to a singly linked list, a doubly linked list maintains references in two directions. Each node contains references (pointer) to both its successor (the next node) and predecessor (the previous node). Although doubly linked lists offer more flexibility for traversing in either direction, they also consume more memory space.</li>
<li><strong>Singly linked list</strong>: This is the standard linked list described earlier. Nodes in a singly linked list include a value and a reference to the next node. The first node is known as the head node, and the last node, which points to null (<code>None</code>), is the tail node.</li>
<li><strong>Circular linked list</strong>: This is formed when the tail node of a singly linked list points back to the head node, creating a loop. In a circular linked list, any node can function as the head node.</li>
<li><strong>Doubly linked list</strong>: In contrast to a singly linked list, a doubly linked list maintains references in two directions. Each node contains references (pointer) to both its successor (the next node) and predecessor (the previous node). Although doubly linked lists offer more flexibility for traversing in either direction, they also consume more memory space.</li>
</ul>
<div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="k">class</span> <span class="nc">ListNode</span><span class="p">:</span>
<a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Bidirectional linked list node class&quot;&quot;&quot;</span><span class="s2">&quot;</span>
<a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Bidirectional linked list node class&quot;&quot;&quot;</span>
<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span> <span class="c1"># Node value</span>
<a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># Reference to the successor node</span>
@@ -3514,26 +3520,26 @@
</div>
</div>
</div>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common Types of Linked Lists" class="animation-figure" src="../linked_list.assets/linkedlist_common_types.png" /></a></p>
<p align="center"> Figure 4-8 &nbsp; Common Types of Linked Lists </p>
<p><a class="glightbox" href="../linked_list.assets/linkedlist_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common types of linked lists" class="animation-figure" src="../linked_list.assets/linkedlist_common_types.png" /></a></p>
<p align="center"> Figure 4-8 &nbsp; Common types of linked lists </p>
<h2 id="424-typical-applications-of-linked-lists">4.2.4 &nbsp; Typical Applications of Linked Lists<a class="headerlink" href="#424-typical-applications-of-linked-lists" title="Permanent link">&para;</a></h2>
<h2 id="424-typical-applications-of-linked-lists">4.2.4 &nbsp; Typical applications of linked lists<a class="headerlink" href="#424-typical-applications-of-linked-lists" title="Permanent link">&para;</a></h2>
<p>Singly linked lists are frequently utilized in implementing stacks, queues, hash tables, and graphs.</p>
<ul>
<li><strong>Stacks and Queues</strong>: In singly linked lists, if insertions and deletions occur at the same end, it behaves like a stack (last-in-first-out). Conversely, if insertions are at one end and deletions at the other, it functions like a queue (first-in-first-out).</li>
<li><strong>Hash Tables</strong>: Linked lists are used in chaining, a popular method for resolving hash collisions. Here, all collided elements are grouped into a linked list.</li>
<li><strong>Stacks and queues</strong>: In singly linked lists, if insertions and deletions occur at the same end, it behaves like a stack (last-in-first-out). Conversely, if insertions are at one end and deletions at the other, it functions like a queue (first-in-first-out).</li>
<li><strong>Hash tables</strong>: Linked lists are used in chaining, a popular method for resolving hash collisions. Here, all collided elements are grouped into a linked list.</li>
<li><strong>Graphs</strong>: Adjacency lists, a standard method for graph representation, associate each graph vertex with a linked list. This list contains elements that represent vertices connected to the corresponding vertex.</li>
</ul>
<p>Doubly linked lists are ideal for scenarios requiring rapid access to preceding and succeeding elements.</p>
<ul>
<li><strong>Advanced Data Structures</strong>: In structures like red-black trees and B-trees, accessing a node's parent is essential. This is achieved by incorporating a reference to the parent node in each node, akin to a doubly linked list.</li>
<li><strong>Browser History</strong>: In web browsers, doubly linked lists facilitate navigating the history of visited pages when users click forward or back.</li>
<li><strong>LRU Algorithm</strong>: Doubly linked lists are apt for Least Recently Used (LRU) cache eviction algorithms, enabling swift identification of the least recently used data and facilitating fast node addition and removal.</li>
<li><strong>Advanced data structures</strong>: In structures like red-black trees and B-trees, accessing a node's parent is essential. This is achieved by incorporating a reference to the parent node in each node, akin to a doubly linked list.</li>
<li><strong>Browser history</strong>: In web browsers, doubly linked lists facilitate navigating the history of visited pages when users click forward or back.</li>
<li><strong>LRU algorithm</strong>: Doubly linked lists are apt for Least Recently Used (LRU) cache eviction algorithms, enabling swift identification of the least recently used data and facilitating fast node addition and removal.</li>
</ul>
<p>Circular linked lists are ideal for applications that require periodic operations, such as resource scheduling in operating systems.</p>
<ul>
<li><strong>Round-Robin Scheduling Algorithm</strong>: In operating systems, the round-robin scheduling algorithm is a common CPU scheduling method, requiring cycling through a group of processes. Each process is assigned a time slice, and upon expiration, the CPU rotates to the next process. This cyclical operation can be efficiently realized using a circular linked list, allowing for a fair and time-shared system among all processes.</li>
<li><strong>Data Buffers</strong>: Circular linked lists are also used in data buffers, like in audio and video players, where the data stream is divided into multiple buffer blocks arranged in a circular fashion for seamless playback.</li>
<li><strong>Round-robin scheduling algorithm</strong>: In operating systems, the round-robin scheduling algorithm is a common CPU scheduling method, requiring cycling through a group of processes. Each process is assigned a time slice, and upon expiration, the CPU rotates to the next process. This cyclical operation can be efficiently realized using a circular linked list, allowing for a fair and time-shared system among all processes.</li>
<li><strong>Data buffers</strong>: Circular linked lists are also used in data buffers, like in audio and video players, where the data stream is divided into multiple buffer blocks arranged in a circular fashion for seamless playback.</li>
</ul>
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en/chapter_array_and_linkedlist/list/index.html
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -998,7 +1004,7 @@
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<a href="#431-common-list-operations" class="md-nav__link">
<span class="md-ellipsis">
4.3.1 &nbsp; Common List Operations
4.3.1 &nbsp; Common list operations
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<nav class="md-nav" aria-label="4.3.1   Common list operations">
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<span class="md-ellipsis">
1. &nbsp; Initializing a List
1. &nbsp; Initializing a list
</span>
</a>
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2. &nbsp; Accessing Elements
2. &nbsp; Accessing elements
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3. &nbsp; Inserting and Removing Elements
3. &nbsp; Inserting and removing elements
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4. &nbsp; Iterating the List
4. &nbsp; Iterating the list
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5. &nbsp; Concatenating Lists
5. &nbsp; Concatenating lists
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6. &nbsp; Sorting the List
6. &nbsp; Sorting the list
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4.3.2 &nbsp; List Implementation
4.3.2 &nbsp; List implementation
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
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<a href="#431-common-list-operations" class="md-nav__link">
<span class="md-ellipsis">
4.3.1 &nbsp; Common List Operations
4.3.1 &nbsp; Common list operations
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1. &nbsp; Initializing a List
1. &nbsp; Initializing a list
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2. &nbsp; Accessing Elements
2. &nbsp; Accessing elements
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3. &nbsp; Inserting and Removing Elements
3. &nbsp; Inserting and removing elements
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<a href="#4-iterating-the-list" class="md-nav__link">
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4. &nbsp; Iterating the List
4. &nbsp; Iterating the list
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<a href="#5-concatenating-lists" class="md-nav__link">
<span class="md-ellipsis">
5. &nbsp; Concatenating Lists
5. &nbsp; Concatenating lists
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<a href="#6-sorting-the-list" class="md-nav__link">
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6. &nbsp; Sorting the List
6. &nbsp; Sorting the list
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4.3.2 &nbsp; List Implementation
4.3.2 &nbsp; List implementation
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<p>When implementing lists using arrays, <strong>the immutability of length reduces the practicality of the list</strong>. This is because predicting the amount of data to be stored in advance is often challenging, making it difficult to choose an appropriate list length. If the length is too small, it may not meet the requirements; if too large, it may waste memory space.</p>
<p>To solve this problem, we can implement lists using a "dynamic array." It inherits the advantages of arrays and can dynamically expand during program execution.</p>
<p>In fact, <strong>many programming languages' standard libraries implement lists using dynamic arrays</strong>, such as Python's <code>list</code>, Java's <code>ArrayList</code>, C++'s <code>vector</code>, and C#'s <code>List</code>. In the following discussion, we will consider "list" and "dynamic array" as synonymous concepts.</p>
<h2 id="431-common-list-operations">4.3.1 &nbsp; Common List Operations<a class="headerlink" href="#431-common-list-operations" title="Permanent link">&para;</a></h2>
<h3 id="1-initializing-a-list">1. &nbsp; Initializing a List<a class="headerlink" href="#1-initializing-a-list" title="Permanent link">&para;</a></h3>
<h2 id="431-common-list-operations">4.3.1 &nbsp; Common list operations<a class="headerlink" href="#431-common-list-operations" title="Permanent link">&para;</a></h2>
<h3 id="1-initializing-a-list">1. &nbsp; Initializing a list<a class="headerlink" href="#1-initializing-a-list" title="Permanent link">&para;</a></h3>
<p>We typically use two initialization methods: "without initial values" and "with initial values".</p>
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<div class="tabbed-content">
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</div>
</div>
<h3 id="2-accessing-elements">2. &nbsp; Accessing Elements<a class="headerlink" href="#2-accessing-elements" title="Permanent link">&para;</a></h3>
<h3 id="2-accessing-elements">2. &nbsp; Accessing elements<a class="headerlink" href="#2-accessing-elements" title="Permanent link">&para;</a></h3>
<p>Lists are essentially arrays, thus they can access and update elements in <span class="arithmatex">\(O(1)\)</span> time, which is very efficient.</p>
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<h3 id="3-inserting-and-removing-elements">3. &nbsp; Inserting and Removing Elements<a class="headerlink" href="#3-inserting-and-removing-elements" title="Permanent link">&para;</a></h3>
<h3 id="3-inserting-and-removing-elements">3. &nbsp; Inserting and removing elements<a class="headerlink" href="#3-inserting-and-removing-elements" title="Permanent link">&para;</a></h3>
<p>Compared to arrays, lists offer more flexibility in adding and removing elements. While adding elements to the end of a list is an <span class="arithmatex">\(O(1)\)</span> operation, the efficiency of inserting and removing elements elsewhere in the list remains the same as in arrays, with a time complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
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@@ -2617,7 +2623,7 @@
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</div>
<h3 id="4-iterating-the-list">4. &nbsp; Iterating the List<a class="headerlink" href="#4-iterating-the-list" title="Permanent link">&para;</a></h3>
<h3 id="4-iterating-the-list">4. &nbsp; Iterating the list<a class="headerlink" href="#4-iterating-the-list" title="Permanent link">&para;</a></h3>
<p>Similar to arrays, lists can be iterated either by using indices or by directly iterating through each element.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Zig</label></div>
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@@ -2782,7 +2788,7 @@
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</div>
<h3 id="5-concatenating-lists">5. &nbsp; Concatenating Lists<a class="headerlink" href="#5-concatenating-lists" title="Permanent link">&para;</a></h3>
<h3 id="5-concatenating-lists">5. &nbsp; Concatenating lists<a class="headerlink" href="#5-concatenating-lists" title="Permanent link">&para;</a></h3>
<p>Given a new list <code>nums1</code>, we can append it to the end of the original list.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Zig</label></div>
<div class="tabbed-content">
@@ -2865,7 +2871,7 @@
</div>
</div>
</div>
<h3 id="6-sorting-the-list">6. &nbsp; Sorting the List<a class="headerlink" href="#6-sorting-the-list" title="Permanent link">&para;</a></h3>
<h3 id="6-sorting-the-list">6. &nbsp; Sorting the list<a class="headerlink" href="#6-sorting-the-list" title="Permanent link">&para;</a></h3>
<p>Once the list is sorted, we can employ algorithms commonly used in array-related algorithm problems, such as "binary search" and "two-pointer" algorithms.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Zig</label></div>
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@@ -2934,13 +2940,13 @@
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</div>
<h2 id="432-list-implementation">4.3.2 &nbsp; List Implementation<a class="headerlink" href="#432-list-implementation" title="Permanent link">&para;</a></h2>
<h2 id="432-list-implementation">4.3.2 &nbsp; List implementation<a class="headerlink" href="#432-list-implementation" title="Permanent link">&para;</a></h2>
<p>Many programming languages come with built-in lists, including Java, C++, Python, etc. Their implementations tend to be intricate, featuring carefully considered settings for various parameters, like initial capacity and expansion factors. Readers who are curious can delve into the source code for further learning.</p>
<p>To enhance our understanding of how lists work, we will attempt to implement a simplified version of a list, focusing on three crucial design aspects:</p>
<ul>
<li><strong>Initial Capacity</strong>: Choose a reasonable initial capacity for the array. In this example, we choose 10 as the initial capacity.</li>
<li><strong>Size Recording</strong>: Declare a variable <code>size</code> to record the current number of elements in the list, updating in real-time with element insertion and deletion. With this variable, we can locate the end of the list and determine whether expansion is needed.</li>
<li><strong>Expansion Mechanism</strong>: If the list reaches full capacity upon an element insertion, an expansion process is required. This involves creating a larger array based on the expansion factor, and then transferring all elements from the current array to the new one. In this example, we stipulate that the array size should double with each expansion.</li>
<li><strong>Initial capacity</strong>: Choose a reasonable initial capacity for the array. In this example, we choose 10 as the initial capacity.</li>
<li><strong>Size recording</strong>: Declare a variable <code>size</code> to record the current number of elements in the list, updating in real-time with element insertion and deletion. With this variable, we can locate the end of the list and determine whether expansion is needed.</li>
<li><strong>Expansion mechanism</strong>: If the list reaches full capacity upon an element insertion, an expansion process is required. This involves creating a larger array based on the expansion factor, and then transferring all elements from the current array to the new one. In this example, we stipulate that the array size should double with each expansion.</li>
</ul>
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<title>4.4 Memory and Cache - Hello Algo</title>
<title>4.4 Memory and cache - Hello Algo</title>
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4.4 Memory and Cache
4.4 Memory and cache
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0.1 About This Book
0.1 About this book
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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4.4 Memory and Cache
4.4 Memory and cache
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4.4.1 &nbsp; Computer Storage Devices
4.4.1 &nbsp; Computer storage devices
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4.4.2 &nbsp; Memory Efficiency of Data Structures
4.4.2 &nbsp; Memory efficiency of data structures
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4.4.3 &nbsp; Cache efficiency of data structures
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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4.4.1 &nbsp; Computer Storage Devices
4.4.1 &nbsp; Computer storage devices
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4.4.2 &nbsp; Memory Efficiency of Data Structures
4.4.2 &nbsp; Memory efficiency of data structures
</span>
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4.4.3 &nbsp; Cache Efficiency of Data Structures
4.4.3 &nbsp; Cache efficiency of data structures
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<!-- Page content -->
<h1 id="44-memory-and-cache">4.4 &nbsp; Memory and Cache *<a class="headerlink" href="#44-memory-and-cache" title="Permanent link">&para;</a></h1>
<h1 id="44-memory-and-cache">4.4 &nbsp; Memory and cache *<a class="headerlink" href="#44-memory-and-cache" title="Permanent link">&para;</a></h1>
<p>In the first two sections of this chapter, we explored arrays and linked lists, two fundamental and important data structures, representing "continuous storage" and "dispersed storage" respectively.</p>
<p>In fact, <strong>the physical structure largely determines the efficiency of a program's use of memory and cache</strong>, which in turn affects the overall performance of the algorithm.</p>
<h2 id="441-computer-storage-devices">4.4.1 &nbsp; Computer Storage Devices<a class="headerlink" href="#441-computer-storage-devices" title="Permanent link">&para;</a></h2>
<h2 id="441-computer-storage-devices">4.4.1 &nbsp; Computer storage devices<a class="headerlink" href="#441-computer-storage-devices" title="Permanent link">&para;</a></h2>
<p>There are three types of storage devices in computers: "hard disk," "random-access memory (RAM)," and "cache memory." The following table shows their different roles and performance characteristics in computer systems.</p>
<p align="center"> Table 4-2 &nbsp; Computer Storage Devices </p>
<p align="center"> Table 4-2 &nbsp; Computer storage devices </p>
<div class="center-table">
<table>
@@ -2144,8 +2150,8 @@
<li><strong>Hard disks are difficult to replace with memory</strong>. Firstly, data in memory is lost after power off, making it unsuitable for long-term data storage; secondly, the cost of memory is dozens of times that of hard disks, making it difficult to popularize in the consumer market.</li>
<li><strong>It is difficult for caches to have both large capacity and high speed</strong>. As the capacity of L1, L2, L3 caches gradually increases, their physical size becomes larger, increasing the physical distance from the CPU core, leading to increased data transfer time and higher element access latency. Under current technology, a multi-level cache structure is the best balance between capacity, speed, and cost.</li>
</ul>
<p><a class="glightbox" href="../ram_and_cache.assets/storage_pyramid.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Computer Storage System" class="animation-figure" src="../ram_and_cache.assets/storage_pyramid.png" /></a></p>
<p align="center"> Figure 4-9 &nbsp; Computer Storage System </p>
<p><a class="glightbox" href="../ram_and_cache.assets/storage_pyramid.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Computer storage system" class="animation-figure" src="../ram_and_cache.assets/storage_pyramid.png" /></a></p>
<p align="center"> Figure 4-9 &nbsp; Computer storage system </p>
<div class="admonition note">
<p class="admonition-title">Note</p>
@@ -2153,29 +2159,29 @@
</div>
<p>Overall, <strong>hard disks are used for long-term storage of large amounts of data, memory is used for temporary storage of data being processed during program execution, and cache is used to store frequently accessed data and instructions</strong> to improve program execution efficiency. Together, they ensure the efficient operation of computer systems.</p>
<p>As shown in the Figure 4-10 , during program execution, data is read from the hard disk into memory for CPU computation. The cache can be considered a part of the CPU, <strong>smartly loading data from memory</strong> to provide fast data access to the CPU, significantly enhancing program execution efficiency and reducing reliance on slower memory.</p>
<p><a class="glightbox" href="../ram_and_cache.assets/computer_storage_devices.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Data Flow Between Hard Disk, Memory, and Cache" class="animation-figure" src="../ram_and_cache.assets/computer_storage_devices.png" /></a></p>
<p align="center"> Figure 4-10 &nbsp; Data Flow Between Hard Disk, Memory, and Cache </p>
<p><a class="glightbox" href="../ram_and_cache.assets/computer_storage_devices.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Data flow between hard disk, memory, and cache" class="animation-figure" src="../ram_and_cache.assets/computer_storage_devices.png" /></a></p>
<p align="center"> Figure 4-10 &nbsp; Data flow between hard disk, memory, and cache </p>
<h2 id="442-memory-efficiency-of-data-structures">4.4.2 &nbsp; Memory Efficiency of Data Structures<a class="headerlink" href="#442-memory-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
<h2 id="442-memory-efficiency-of-data-structures">4.4.2 &nbsp; Memory efficiency of data structures<a class="headerlink" href="#442-memory-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
<p>In terms of memory space utilization, arrays and linked lists have their advantages and limitations.</p>
<p>On one hand, <strong>memory is limited and cannot be shared by multiple programs</strong>, so we hope that data structures can use space as efficiently as possible. The elements of an array are tightly packed without extra space for storing references (pointers) between linked list nodes, making them more space-efficient. However, arrays require allocating sufficient continuous memory space at once, which may lead to memory waste, and array expansion also requires additional time and space costs. In contrast, linked lists allocate and reclaim memory dynamically on a per-node basis, providing greater flexibility.</p>
<p>On the other hand, during program execution, <strong>as memory is repeatedly allocated and released, the degree of fragmentation of free memory becomes higher</strong>, leading to reduced memory utilization efficiency. Arrays, due to their continuous storage method, are relatively less likely to cause memory fragmentation. In contrast, the elements of a linked list are dispersedly stored, and frequent insertion and deletion operations make memory fragmentation more likely.</p>
<h2 id="443-cache-efficiency-of-data-structures">4.4.3 &nbsp; Cache Efficiency of Data Structures<a class="headerlink" href="#443-cache-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
<h2 id="443-cache-efficiency-of-data-structures">4.4.3 &nbsp; Cache efficiency of data structures<a class="headerlink" href="#443-cache-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
<p>Although caches are much smaller in space capacity than memory, they are much faster and play a crucial role in program execution speed. Since the cache's capacity is limited and can only store a small part of frequently accessed data, when the CPU tries to access data not in the cache, a "cache miss" occurs, forcing the CPU to load the needed data from slower memory.</p>
<p>Clearly, <strong>the fewer the cache misses, the higher the CPU's data read-write efficiency</strong>, and the better the program performance. The proportion of successful data retrieval from the cache by the CPU is called the "cache hit rate," a metric often used to measure cache efficiency.</p>
<p>To achieve higher efficiency, caches adopt the following data loading mechanisms.</p>
<ul>
<li><strong>Cache Lines</strong>: Caches don't store and load data byte by byte but in units of cache lines. Compared to byte-by-byte transfer, the transmission of cache lines is more efficient.</li>
<li><strong>Prefetch Mechanism</strong>: Processors try to predict data access patterns (such as sequential access, fixed stride jumping access, etc.) and load data into the cache according to specific patterns to improve the hit rate.</li>
<li><strong>Spatial Locality</strong>: If data is accessed, data nearby is likely to be accessed in the near future. Therefore, when loading certain data, the cache also loads nearby data to improve the hit rate.</li>
<li><strong>Temporal Locality</strong>: If data is accessed, it's likely to be accessed again in the near future. Caches use this principle to retain recently accessed data to improve the hit rate.</li>
<li><strong>Cache lines</strong>: Caches don't store and load data byte by byte but in units of cache lines. Compared to byte-by-byte transfer, the transmission of cache lines is more efficient.</li>
<li><strong>Prefetch mechanism</strong>: Processors try to predict data access patterns (such as sequential access, fixed stride jumping access, etc.) and load data into the cache according to specific patterns to improve the hit rate.</li>
<li><strong>Spatial locality</strong>: If data is accessed, data nearby is likely to be accessed in the near future. Therefore, when loading certain data, the cache also loads nearby data to improve the hit rate.</li>
<li><strong>Temporal locality</strong>: If data is accessed, it's likely to be accessed again in the near future. Caches use this principle to retain recently accessed data to improve the hit rate.</li>
</ul>
<p>In fact, <strong>arrays and linked lists have different cache utilization efficiencies</strong>, mainly reflected in the following aspects.</p>
<ul>
<li><strong>Occupied Space</strong>: Linked list elements occupy more space than array elements, resulting in less effective data volume in the cache.</li>
<li><strong>Cache Lines</strong>: Linked list data is scattered throughout memory, and since caches load "by line," the proportion of loading invalid data is higher.</li>
<li><strong>Prefetch Mechanism</strong>: The data access pattern of arrays is more "predictable" than that of linked lists, meaning the system is more likely to guess which data will be loaded next.</li>
<li><strong>Spatial Locality</strong>: Arrays are stored in concentrated memory spaces, so the data near the loaded data is more likely to be accessed next.</li>
<li><strong>Occupied space</strong>: Linked list elements occupy more space than array elements, resulting in less effective data volume in the cache.</li>
<li><strong>Cache lines</strong>: Linked list data is scattered throughout memory, and since caches load "by line," the proportion of loading invalid data is higher.</li>
<li><strong>Prefetch mechanism</strong>: The data access pattern of arrays is more "predictable" than that of linked lists, meaning the system is more likely to guess which data will be loaded next.</li>
<li><strong>Spatial locality</strong>: Arrays are stored in concentrated memory spaces, so the data near the loaded data is more likely to be accessed next.</li>
</ul>
<p>Overall, <strong>arrays have a higher cache hit rate and are generally more efficient in operation than linked lists</strong>. This makes data structures based on arrays more popular in solving algorithmic problems.</p>
<p>It should be noted that <strong>high cache efficiency does not mean that arrays are always better than linked lists</strong>. Which data structure to choose in actual applications should be based on specific requirements. For example, both arrays and linked lists can implement the "stack" data structure (which will be detailed in the next chapter), but they are suitable for different scenarios.</p>
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
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1.1 Algorithms are Everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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3.1 Classification of Data Structures
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Chapter 4. Array and Linked List
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Chapter 5. Stack and Queue
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<!-- Page content -->
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<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
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<ul>
<li>Arrays and linked lists are two basic data structures, representing two storage methods in computer memory: contiguous space storage and non-contiguous space storage. Their characteristics complement each other.</li>
<li>Arrays support random access and use less memory; however, they are inefficient in inserting and deleting elements and have a fixed length after initialization.</li>
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<p><strong>Q</strong>: Why do arrays require elements of the same type, while linked lists do not emphasize same-type elements?</p>
<p>Linked lists consist of nodes connected by references (pointers), and each node can store data of different types, such as int, double, string, object, etc.</p>
<p>In contrast, array elements must be of the same type, allowing the calculation of offsets to access the corresponding element positions. For example, an array containing both int and long types, with single elements occupying 4 bytes and 8 bytes respectively, cannot use the following formula to calculate offsets, as the array contains elements of two different lengths.</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Element memory address = Array memory address + Element length * Element index</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Element memory address = array memory address + element length * element index</span>
</code></pre></div>
<p><strong>Q</strong>: After deleting a node, is it necessary to set <code>P.next</code> to <code>None</code>?</p>
<p>Not modifying <code>P.next</code> is also acceptable. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been effectively removed from the list, and where <code>P</code> points no longer affects the list.</p>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1327,7 +1333,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
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@@ -1344,7 +1350,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2003,8 +2009,8 @@
<!-- Page content -->
<h1 id="chapter-2-complexity-analysis">Chapter 2. &nbsp; Complexity Analysis<a class="headerlink" href="#chapter-2-complexity-analysis" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_complexity_analysis.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="complexity_analysis" class="cover-image" src="../assets/covers/chapter_complexity_analysis.jpg" /></a></p>
<h1 id="chapter-2-complexity-analysis">Chapter 2. &nbsp; Complexity analysis<a class="headerlink" href="#chapter-2-complexity-analysis" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_complexity_analysis.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Complexity analysis" class="cover-image" src="../assets/covers/chapter_complexity_analysis.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Complexity analysis is like a space-time navigator in the vast universe of algorithms.</p>
@@ -2012,10 +2018,10 @@
</div>
<h2 id="chapter-contents">Chapter Contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/performance_evaluation/">2.1 &nbsp; Algorithm Efficiency Assessment</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/iteration_and_recursion/">2.2 &nbsp; Iteration and Recursion</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/time_complexity/">2.3 &nbsp; Time Complexity</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/space_complexity/">2.4 &nbsp; Space Complexity</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/performance_evaluation/">2.1 &nbsp; Algorithm efficiency assessment</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/iteration_and_recursion/">2.2 &nbsp; Iteration and recursion</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/time_complexity/">2.3 &nbsp; Time complexity</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/space_complexity/">2.4 &nbsp; Space complexity</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_computational_complexity/summary/">2.5 &nbsp; Summary</a></li>
</ul>
@@ -2064,7 +2070,7 @@ aria-label="Footer"
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class="md-footer__link md-footer__link--next"
aria-label="Next: 2.1 Algorithm Efficiency Assessment"
aria-label="Next: 2.1 Algorithm efficiency assessment"
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Next
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.1 Algorithm efficiency assessment
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en/chapter_computational_complexity/iteration_and_recursion/index.html
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<title>2.2 Iteration and Recursion - Hello Algo</title>
<title>2.2 Iteration and recursion - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
</div>
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -628,7 +634,7 @@
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<li class="md-nav__item">
<a href="#1-for-loops" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; For Loops
1. &nbsp; For loops
</span>
</a>
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<a href="#2-while-loops" class="md-nav__link">
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2. &nbsp; While Loops
2. &nbsp; While loops
</span>
</a>
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<li class="md-nav__item">
<a href="#3-nested-loops" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Nested Loops
3. &nbsp; Nested loops
</span>
</a>
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<li class="md-nav__item">
<a href="#1-call-stack" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Call Stack
1. &nbsp; Call stack
</span>
</a>
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<a href="#2-tail-recursion" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Tail Recursion
2. &nbsp; Tail recursion
</span>
</a>
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<li class="md-nav__item">
<a href="#3-recursion-tree" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Recursion Tree
3. &nbsp; Recursion tree
</span>
</a>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -938,7 +944,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -954,7 +960,7 @@
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<label class="md-nav__title" for="__nav_5">
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1306,7 +1312,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1460,7 +1466,7 @@
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1477,7 +1483,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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<li class="md-nav__item">
<a href="#1-for-loops" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; For Loops
1. &nbsp; For loops
</span>
</a>
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<a href="#2-while-loops" class="md-nav__link">
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2. &nbsp; While Loops
2. &nbsp; While loops
</span>
</a>
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<li class="md-nav__item">
<a href="#3-nested-loops" class="md-nav__link">
<span class="md-ellipsis">
3. &nbsp; Nested Loops
3. &nbsp; Nested loops
</span>
</a>
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<li class="md-nav__item">
<a href="#1-call-stack" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Call Stack
1. &nbsp; Call stack
</span>
</a>
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<a href="#2-tail-recursion" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Tail Recursion
2. &nbsp; Tail recursion
</span>
</a>
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<a href="#3-recursion-tree" class="md-nav__link">
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3. &nbsp; Recursion Tree
3. &nbsp; Recursion tree
</span>
</a>
@@ -2220,11 +2226,11 @@
<!-- Page content -->
<h1 id="22-iteration-and-recursion">2.2 &nbsp; Iteration and Recursion<a class="headerlink" href="#22-iteration-and-recursion" title="Permanent link">&para;</a></h1>
<h1 id="22-iteration-and-recursion">2.2 &nbsp; Iteration and recursion<a class="headerlink" href="#22-iteration-and-recursion" title="Permanent link">&para;</a></h1>
<p>In algorithms, the repeated execution of a task is quite common and is closely related to the analysis of complexity. Therefore, before delving into the concepts of time complexity and space complexity, let's first explore how to implement repetitive tasks in programming. This involves understanding two fundamental programming control structures: iteration and recursion.</p>
<h2 id="221-iteration">2.2.1 &nbsp; Iteration<a class="headerlink" href="#221-iteration" title="Permanent link">&para;</a></h2>
<p>"Iteration" is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met until this condition is no longer satisfied.</p>
<h3 id="1-for-loops">1. &nbsp; For Loops<a class="headerlink" href="#1-for-loops" title="Permanent link">&para;</a></h3>
<h3 id="1-for-loops">1. &nbsp; For loops<a class="headerlink" href="#1-for-loops" title="Permanent link">&para;</a></h3>
<p>The <code>for</code> loop is one of the most common forms of iteration, and <strong>it's particularly suitable when the number of iterations is known in advance</strong>.</p>
<p>The following function uses a <code>for</code> loop to perform a summation of <span class="arithmatex">\(1 + 2 + \dots + n\)</span>, with the sum being stored in the variable <code>res</code>. It's important to note that in Python, <code>range(a, b)</code> creates an interval that is inclusive of <code>a</code> but exclusive of <code>b</code>, meaning it iterates over the range from <span class="arithmatex">\(a\)</span> up to <span class="arithmatex">\(b−1\)</span>.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:14"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label><label for="__tabbed_1_14">Zig</label></div>
@@ -2405,11 +2411,11 @@
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20for_loop%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22for%20%E5%BE%AA%E7%8E%AF%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E6%B1%82%E5%92%8C%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20for_loop%28n%29%0A%20%20%20%20print%28f%22%5Cnfor%20%E5%BE%AA%E7%8E%AF%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>The flowchart below represents this sum function.</p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Flowchart of the Sum Function" class="animation-figure" src="../iteration_and_recursion.assets/iteration.png" /></a></p>
<p align="center"> Figure 2-1 &nbsp; Flowchart of the Sum Function </p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Flowchart of the sum function" class="animation-figure" src="../iteration_and_recursion.assets/iteration.png" /></a></p>
<p align="center"> Figure 2-1 &nbsp; Flowchart of the sum function </p>
<p>The number of operations in this summation function is proportional to the size of the input data <span class="arithmatex">\(n\)</span>, or in other words, it has a "linear relationship." This "linear relationship" is what time complexity describes. This topic will be discussed in more detail in the next section.</p>
<h3 id="2-while-loops">2. &nbsp; While Loops<a class="headerlink" href="#2-while-loops" title="Permanent link">&para;</a></h3>
<h3 id="2-while-loops">2. &nbsp; While loops<a class="headerlink" href="#2-while-loops" title="Permanent link">&para;</a></h3>
<p>Similar to <code>for</code> loops, <code>while</code> loops are another approach for implementing iteration. In a <code>while</code> loop, the program checks a condition at the beginning of each iteration; if the condition is true, the execution continues, otherwise, the loop ends.</p>
<p>Below we use a <code>while</code> loop to implement the sum <span class="arithmatex">\(1 + 2 + \dots + n\)</span>.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:14"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label><label for="__tabbed_2_14">Zig</label></div>
@@ -2858,7 +2864,7 @@
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20while_loop_ii%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22while%20%E5%BE%AA%E7%8E%AF%EF%BC%88%E4%B8%A4%E6%AC%A1%E6%9B%B4%E6%96%B0%EF%BC%89%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20i%20%3D%201%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%9D%A1%E4%BB%B6%E5%8F%98%E9%87%8F%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E6%B1%82%E5%92%8C%201,%204,%2010,%20...%0A%20%20%20%20while%20i%20%3C%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20%20%20%20%20%23%20%E6%9B%B4%E6%96%B0%E6%9D%A1%E4%BB%B6%E5%8F%98%E9%87%8F%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20i%20*%3D%202%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20while_loop_ii%28n%29%0A%20%20%20%20print%28f%22%5Cnwhile%20%E5%BE%AA%E7%8E%AF%EF%BC%88%E4%B8%A4%E6%AC%A1%E6%9B%B4%E6%96%B0%EF%BC%89%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>Overall, <strong><code>for</code> loops are more concise, while <code>while</code> loops are more flexible</strong>. Both can implement iterative structures. Which one to use should be determined based on the specific requirements of the problem.</p>
<h3 id="3-nested-loops">3. &nbsp; Nested Loops<a class="headerlink" href="#3-nested-loops" title="Permanent link">&para;</a></h3>
<h3 id="3-nested-loops">3. &nbsp; Nested loops<a class="headerlink" href="#3-nested-loops" title="Permanent link">&para;</a></h3>
<p>We can nest one loop structure within another. Below is an example using <code>for</code> loops:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label><label for="__tabbed_4_14">Zig</label></div>
<div class="tabbed-content">
@@ -3086,8 +3092,8 @@
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20nested_for_loop%28n%3A%20int%29%20-%3E%20str%3A%0A%20%20%20%20%22%22%22%E5%8F%8C%E5%B1%82%20for%20%E5%BE%AA%E7%8E%AF%22%22%22%0A%20%20%20%20res%20%3D%20%22%22%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%20i%20%3D%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%20j%20%3D%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20f%22%28%7Bi%7D,%20%7Bj%7D%29,%20%22%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20nested_for_loop%28n%29%0A%20%20%20%20print%28f%22%5Cn%E5%8F%8C%E5%B1%82%20for%20%E5%BE%AA%E7%8E%AF%E7%9A%84%E9%81%8D%E5%8E%86%E7%BB%93%E6%9E%9C%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>The flowchart below represents this nested loop.</p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/nested_iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Flowchart of the Nested Loop" class="animation-figure" src="../iteration_and_recursion.assets/nested_iteration.png" /></a></p>
<p align="center"> Figure 2-2 &nbsp; Flowchart of the Nested Loop </p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/nested_iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Flowchart of the nested loop" class="animation-figure" src="../iteration_and_recursion.assets/nested_iteration.png" /></a></p>
<p align="center"> Figure 2-2 &nbsp; Flowchart of the nested loop </p>
<p>In such cases, the number of operations of the function is proportional to <span class="arithmatex">\(n^2\)</span>, meaning the algorithm's runtime and the size of the input data <span class="arithmatex">\(n\)</span> has a 'quadratic relationship.'</p>
<p>We can further increase the complexity by adding more nested loops, each level of nesting effectively "increasing the dimension," which raises the time complexity to "cubic," "quartic," and so on.</p>
@@ -3295,8 +3301,8 @@
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20%E9%80%92%EF%BC%9A%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%0A%20%20%20%20res%20%3D%20recur%28n%20-%201%29%0A%20%20%20%20%23%20%E5%BD%92%EF%BC%9A%E8%BF%94%E5%9B%9E%E7%BB%93%E6%9E%9C%0A%20%20%20%20return%20n%20%2B%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20recur%28n%29%0A%20%20%20%20print%28f%22%5Cn%E9%80%92%E5%BD%92%E5%87%BD%E6%95%B0%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>The Figure 2-3 shows the recursive process of this function.</p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_sum.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursive Process of the Sum Function" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum.png" /></a></p>
<p align="center"> Figure 2-3 &nbsp; Recursive Process of the Sum Function </p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_sum.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursive process of the sum function" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum.png" /></a></p>
<p align="center"> Figure 2-3 &nbsp; Recursive process of the sum function </p>
<p>Although iteration and recursion can achieve the same results from a computational standpoint, <strong>they represent two entirely different paradigms of thinking and problem-solving</strong>.</p>
<ul>
@@ -3308,22 +3314,22 @@
<li><strong>Iteration</strong>: In this approach, we simulate the summation process within a loop. Starting from <span class="arithmatex">\(1\)</span> and traversing to <span class="arithmatex">\(n\)</span>, we perform the summation operation in each iteration to eventually compute <span class="arithmatex">\(f(n)\)</span>.</li>
<li><strong>Recursion</strong>: Here, the problem is broken down into a sub-problem: <span class="arithmatex">\(f(n) = n + f(n-1)\)</span>. This decomposition continues recursively until reaching the base case, <span class="arithmatex">\(f(1) = 1\)</span>, at which point the recursion terminates.</li>
</ul>
<h3 id="1-call-stack">1. &nbsp; Call Stack<a class="headerlink" href="#1-call-stack" title="Permanent link">&para;</a></h3>
<h3 id="1-call-stack">1. &nbsp; Call stack<a class="headerlink" href="#1-call-stack" title="Permanent link">&para;</a></h3>
<p>Every time a recursive function calls itself, the system allocates memory for the newly initiated function to store local variables, the return address, and other relevant information. This leads to two primary outcomes.</p>
<ul>
<li>The function's context data is stored in a memory area called "stack frame space" and is only released after the function returns. Therefore, <strong>recursion generally consumes more memory space than iteration</strong>.</li>
<li>Recursive calls introduce additional overhead. <strong>Hence, recursion is usually less time-efficient than loops.</strong></li>
</ul>
<p>As shown in the Figure 2-4 , there are <span class="arithmatex">\(n\)</span> unreturned recursive functions before triggering the termination condition, indicating a <strong>recursion depth of <span class="arithmatex">\(n\)</span></strong>.</p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_sum_depth.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursion Call Depth" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum_depth.png" /></a></p>
<p align="center"> Figure 2-4 &nbsp; Recursion Call Depth </p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_sum_depth.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursion call depth" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum_depth.png" /></a></p>
<p align="center"> Figure 2-4 &nbsp; Recursion call depth </p>
<p>In practice, the depth of recursion allowed by programming languages is usually limited, and excessively deep recursion can lead to stack overflow errors.</p>
<h3 id="2-tail-recursion">2. &nbsp; Tail Recursion<a class="headerlink" href="#2-tail-recursion" title="Permanent link">&para;</a></h3>
<h3 id="2-tail-recursion">2. &nbsp; Tail recursion<a class="headerlink" href="#2-tail-recursion" title="Permanent link">&para;</a></h3>
<p>Interestingly, <strong>if a function performs its recursive call as the very last step before returning,</strong> it can be optimized by the compiler or interpreter to be as space-efficient as iteration. This scenario is known as "tail recursion."</p>
<ul>
<li><strong>Regular Recursion</strong>: In standard recursion, when the function returns to the previous level, it continues to execute more code, requiring the system to save the context of the previous call.</li>
<li><strong>Tail Recursion</strong>: Here, the recursive call is the final operation before the function returns. This means that upon returning to the previous level, no further actions are needed, so the system does not need to save the context of the previous level.</li>
<li><strong>Regular recursion</strong>: In standard recursion, when the function returns to the previous level, it continues to execute more code, requiring the system to save the context of the previous call.</li>
<li><strong>Tail recursion</strong>: Here, the recursive call is the final operation before the function returns. This means that upon returning to the previous level, no further actions are needed, so the system does not need to save the context of the previous level.</li>
</ul>
<p>For example, in calculating <span class="arithmatex">\(1 + 2 + \dots + n\)</span>, we can make the result variable <code>res</code> a parameter of the function, thereby achieving tail recursion:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="6:14"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><input id="__tabbed_6_14" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label><label for="__tabbed_6_14">Zig</label></div>
@@ -3491,17 +3497,17 @@
</details>
<p>The execution process of tail recursion is shown in the following figure. Comparing regular recursion and tail recursion, the point of the summation operation is different.</p>
<ul>
<li><strong>Regular Recursion</strong>: The summation operation occurs during the "returning" phase, requiring another summation after each layer returns.</li>
<li><strong>Tail Recursion</strong>: The summation operation occurs during the "calling" phase, and the "returning" phase only involves returning through each layer.</li>
<li><strong>Regular recursion</strong>: The summation operation occurs during the "returning" phase, requiring another summation after each layer returns.</li>
<li><strong>Tail recursion</strong>: The summation operation occurs during the "calling" phase, and the "returning" phase only involves returning through each layer.</li>
</ul>
<p><a class="glightbox" href="../iteration_and_recursion.assets/tail_recursion_sum.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Tail Recursion Process" class="animation-figure" src="../iteration_and_recursion.assets/tail_recursion_sum.png" /></a></p>
<p align="center"> Figure 2-5 &nbsp; Tail Recursion Process </p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/tail_recursion_sum.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Tail recursion process" class="animation-figure" src="../iteration_and_recursion.assets/tail_recursion_sum.png" /></a></p>
<p align="center"> Figure 2-5 &nbsp; Tail recursion process </p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if the function is in the form of tail recursion, it may still encounter stack overflow issues.</p>
</div>
<h3 id="3-recursion-tree">3. &nbsp; Recursion Tree<a class="headerlink" href="#3-recursion-tree" title="Permanent link">&para;</a></h3>
<h3 id="3-recursion-tree">3. &nbsp; Recursion tree<a class="headerlink" href="#3-recursion-tree" title="Permanent link">&para;</a></h3>
<p>When dealing with algorithms related to "divide and conquer", recursion often offers a more intuitive approach and more readable code than iteration. Take the "Fibonacci sequence" as an example.</p>
<div class="admonition question">
<p class="admonition-title">Question</p>
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20fib%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%20f%281%29%20%3D%200,%20f%282%29%20%3D%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%20-%201%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%20f%28n%29%20%3D%20f%28n-1%29%20%2B%20f%28n-2%29%0A%20%20%20%20res%20%3D%20fib%28n%20-%201%29%20%2B%20fib%28n%20-%202%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E7%BB%93%E6%9E%9C%20f%28n%29%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20fib%28n%29%0A%20%20%20%20print%28f%22%5Cn%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%E7%9A%84%E7%AC%AC%20%7Bn%7D%20%E9%A1%B9%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>Observing the above code, we see that it recursively calls two functions within itself, <strong>meaning that one call generates two branching calls</strong>. As illustrated below, this continuous recursive calling eventually creates a "recursion tree" with a depth of <span class="arithmatex">\(n\)</span>.</p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Fibonacci Sequence Recursion Tree" class="animation-figure" src="../iteration_and_recursion.assets/recursion_tree.png" /></a></p>
<p align="center"> Figure 2-6 &nbsp; Fibonacci Sequence Recursion Tree </p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Fibonacci sequence recursion tree" class="animation-figure" src="../iteration_and_recursion.assets/recursion_tree.png" /></a></p>
<p align="center"> Figure 2-6 &nbsp; Fibonacci sequence recursion tree </p>
<p>Fundamentally, recursion embodies the paradigm of "breaking down a problem into smaller sub-problems." This divide-and-conquer strategy is crucial.</p>
<ul>
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<h2 id="223-comparison">2.2.3 &nbsp; Comparison<a class="headerlink" href="#223-comparison" title="Permanent link">&para;</a></h2>
<p>Summarizing the above content, the following table shows the differences between iteration and recursion in terms of implementation, performance, and applicability.</p>
<p align="center"> Table: Comparison of Iteration and Recursion Characteristics </p>
<p align="center"> Table: Comparison of iteration and recursion characteristics </p>
<div class="center-table">
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
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2.1.2 &nbsp; Theoretical Estimation
2.1.2 &nbsp; Theoretical estimation
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number encoding *
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3.4 Character encoding *
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4.2 Linked list
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4.4 Memory and cache
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Chapter 5. Stack and queue
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Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.3 Hash Algorithm
6.3 Hash algorithm
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2.1.2 &nbsp; Theoretical Estimation
2.1.2 &nbsp; Theoretical estimation
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<h1 id="21-algorithm-efficiency-assessment">2.1 &nbsp; Algorithm Efficiency Assessment<a class="headerlink" href="#21-algorithm-efficiency-assessment" title="Permanent link">&para;</a></h1>
<h1 id="21-algorithm-efficiency-assessment">2.1 &nbsp; Algorithm efficiency assessment<a class="headerlink" href="#21-algorithm-efficiency-assessment" title="Permanent link">&para;</a></h1>
<p>In algorithm design, we pursue the following two objectives in sequence.</p>
<ol>
<li><strong>Finding a Solution to the Problem</strong>: The algorithm should reliably find the correct solution within the stipulated range of inputs.</li>
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</ol>
<p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the merits of an algorithm, which includes the following two dimensions.</p>
<ul>
<li><strong>Time Efficiency</strong>: The speed at which an algorithm runs.</li>
<li><strong>Space Efficiency</strong>: The size of the memory space occupied by an algorithm.</li>
<li><strong>Time efficiency</strong>: The speed at which an algorithm runs.</li>
<li><strong>Space efficiency</strong>: The size of the memory space occupied by an algorithm.</li>
</ul>
<p>In short, <strong>our goal is to design data structures and algorithms that are both fast and memory-efficient</strong>. Effectively assessing algorithm efficiency is crucial because only then can we compare various algorithms and guide the process of algorithm design and optimization.</p>
<p>There are mainly two methods of efficiency assessment: actual testing and theoretical estimation.</p>
<h2 id="211-actual-testing">2.1.1 &nbsp; Actual Testing<a class="headerlink" href="#211-actual-testing" title="Permanent link">&para;</a></h2>
<h2 id="211-actual-testing">2.1.1 &nbsp; Actual testing<a class="headerlink" href="#211-actual-testing" title="Permanent link">&para;</a></h2>
<p>Suppose we have algorithms <code>A</code> and <code>B</code>, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms and monitor and record their runtime and memory usage. This assessment method reflects the actual situation but has significant limitations.</p>
<p>On one hand, <strong>it's difficult to eliminate interference from the testing environment</strong>. Hardware configurations can affect algorithm performance. For example, algorithm <code>A</code> might run faster than <code>B</code> on one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.</p>
<p>On the other hand, <strong>conducting a full test is very resource-intensive</strong>. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm <code>A</code> might run faster than <code>B</code>, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.</p>
<h2 id="212-theoretical-estimation">2.1.2 &nbsp; Theoretical Estimation<a class="headerlink" href="#212-theoretical-estimation" title="Permanent link">&para;</a></h2>
<h2 id="212-theoretical-estimation">2.1.2 &nbsp; Theoretical estimation<a class="headerlink" href="#212-theoretical-estimation" title="Permanent link">&para;</a></h2>
<p>Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as "asymptotic complexity analysis," or simply "complexity analysis."</p>
<p>Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. <strong>It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases</strong>. This definition might sound complex, but we can break it down into three key points to understand it better.</p>
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<title>2.4 Space Complexity - Hello Algo</title>
<title>2.4 Space complexity - Hello Algo</title>
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2.4 Space Complexity
2.4 Space complexity
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中文
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繁體中文
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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2.4 Space Complexity
2.4 Space complexity
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2.4.1 &nbsp; Space Related to Algorithms
2.4.1 &nbsp; Space related to algorithms
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2.4.2 &nbsp; Calculation Method
2.4.2 &nbsp; Calculation method
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2.4.3 &nbsp; Common Types
2.4.3 &nbsp; Common types
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1. &nbsp; Constant Order
1. &nbsp; Constant order
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2. &nbsp; Linear Order
2. &nbsp; Linear order
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3. &nbsp; Quadratic Order
3. &nbsp; Quadratic order
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4. &nbsp; Exponential Order
4. &nbsp; Exponential order
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5. &nbsp; Logarithmic Order
5. &nbsp; Logarithmic order
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2.4.4 &nbsp; Balancing Time and Space
2.4.4 &nbsp; Balancing time and space
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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2.4.1 &nbsp; Space Related to Algorithms
2.4.1 &nbsp; Space related to algorithms
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2.4.2 &nbsp; Calculation Method
2.4.2 &nbsp; Calculation method
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2.4.3 &nbsp; Common Types
2.4.3 &nbsp; Common types
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1. &nbsp; Constant Order
1. &nbsp; Constant order
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2. &nbsp; Linear Order
2. &nbsp; Linear order
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3. &nbsp; Quadratic Order
3. &nbsp; Quadratic order
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4. &nbsp; Exponential Order
4. &nbsp; Exponential order
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5. &nbsp; Logarithmic Order
5. &nbsp; Logarithmic order
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2.4.4 &nbsp; Balancing Time and Space
2.4.4 &nbsp; Balancing time and space
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<!-- Page content -->
<h1 id="24-space-complexity">2.4 &nbsp; Space Complexity<a class="headerlink" href="#24-space-complexity" title="Permanent link">&para;</a></h1>
<h1 id="24-space-complexity">2.4 &nbsp; Space complexity<a class="headerlink" href="#24-space-complexity" title="Permanent link">&para;</a></h1>
<p>"Space complexity" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".</p>
<h2 id="241-space-related-to-algorithms">2.4.1 &nbsp; Space Related to Algorithms<a class="headerlink" href="#241-space-related-to-algorithms" title="Permanent link">&para;</a></h2>
<h2 id="241-space-related-to-algorithms">2.4.1 &nbsp; Space related to algorithms<a class="headerlink" href="#241-space-related-to-algorithms" title="Permanent link">&para;</a></h2>
<p>The memory space used by an algorithm during its execution mainly includes the following types.</p>
<ul>
<li><strong>Input Space</strong>: Used to store the input data of the algorithm.</li>
<li><strong>Temporary Space</strong>: Used to store variables, objects, function contexts, and other data during the algorithm's execution.</li>
<li><strong>Output Space</strong>: Used to store the output data of the algorithm.</li>
<li><strong>Input space</strong>: Used to store the input data of the algorithm.</li>
<li><strong>Temporary space</strong>: Used to store variables, objects, function contexts, and other data during the algorithm's execution.</li>
<li><strong>Output space</strong>: Used to store the output data of the algorithm.</li>
</ul>
<p>Generally, the scope of space complexity statistics includes both "Temporary Space" and "Output Space".</p>
<p>Temporary space can be further divided into three parts.</p>
<ul>
<li><strong>Temporary Data</strong>: Used to save various constants, variables, objects, etc., during the algorithm's execution.</li>
<li><strong>Stack Frame Space</strong>: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li>
<li><strong>Instruction Space</strong>: Used to store compiled program instructions, which are usually negligible in actual statistics.</li>
<li><strong>Temporary data</strong>: Used to save various constants, variables, objects, etc., during the algorithm's execution.</li>
<li><strong>Stack frame space</strong>: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li>
<li><strong>Instruction space</strong>: Used to store compiled program instructions, which are usually negligible in actual statistics.</li>
</ul>
<p>When analyzing the space complexity of a program, <strong>we typically count the Temporary Data, Stack Frame Space, and Output Data</strong>, as shown in the Figure 2-15 .</p>
<p><a class="glightbox" href="../space_complexity.assets/space_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Space Types Used in Algorithms" class="animation-figure" src="../space_complexity.assets/space_types.png" /></a></p>
<p align="center"> Figure 2-15 &nbsp; Space Types Used in Algorithms </p>
<p><a class="glightbox" href="../space_complexity.assets/space_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Space types used in algorithms" class="animation-figure" src="../space_complexity.assets/space_types.png" /></a></p>
<p align="center"> Figure 2-15 &nbsp; Space types used in algorithms </p>
<p>The relevant code is as follows:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span> <span class="nc">Node</span><span class="p">:</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Classes&quot;&quot;&quot;</span><span class="s2">&quot;</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Classes&quot;&quot;&quot;</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">x</span> <span class="c1"># node value</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Node</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># reference to the next node</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="k">def</span> <span class="nf">function</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;&quot;Functions&quot;&quot;&quot;</span><span class="s2">&quot;&quot;</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Functions&quot;&quot;&quot;</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="c1"># Perform certain operations...</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="k">return</span> <span class="mi">0</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
@@ -2458,7 +2464,7 @@
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="n">next</span>: <span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">Node</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="p">}</span>
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a>
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Creating a Node structure */</span>
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Constructor */</span>
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="k">impl</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w"> </span><span class="k">fn</span> <span class="nf">new</span><span class="p">(</span><span class="n">val</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="nc">Self</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w"> </span><span class="bp">Self</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span>: <span class="nc">val</span><span class="p">,</span><span class="w"> </span><span class="n">next</span>: <span class="nb">None</span> <span class="p">}</span>
@@ -2505,7 +2511,7 @@
</div>
</div>
</div>
<h2 id="242-calculation-method">2.4.2 &nbsp; Calculation Method<a class="headerlink" href="#242-calculation-method" title="Permanent link">&para;</a></h2>
<h2 id="242-calculation-method">2.4.2 &nbsp; Calculation method<a class="headerlink" href="#242-calculation-method" title="Permanent link">&para;</a></h2>
<p>The method for calculating space complexity is roughly similar to that of time complexity, with the only change being the shift of the statistical object from "number of operations" to "size of used space".</p>
<p>However, unlike time complexity, <strong>we usually only focus on the worst-case space complexity</strong>. This is because memory space is a hard requirement, and we must ensure that there is enough memory space reserved under all input data.</p>
<p>Consider the following code, the term "worst-case" in worst-case space complexity has two meanings.</p>
@@ -2605,10 +2611,10 @@
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="k">fn</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// O(1)</span>
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">10000</span><span class="p">];</span><span class="w"> </span><span class="c1">// O(1)</span>
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// O(1)</span>
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">10000</span><span class="p">];</span><span class="w"> </span><span class="c1">// O(1)</span>
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">10</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span><span class="w"> </span><span class="c1">// O(n)</span>
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span><span class="w"> </span><span class="c1">// O(n)</span>
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="p">}</span>
</code></pre></div>
@@ -2641,12 +2647,12 @@
<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a> <span class="k">return</span> <span class="mi">0</span>
<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>
<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="k">def</span> <span class="nf">loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Loop O(1)&quot;&quot;&quot;</span><span class="s2">&quot;&quot;</span>
<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Loop O(1)&quot;&quot;&quot;</span>
<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a> <span class="n">function</span><span class="p">()</span>
<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>
<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a><span class="k">def</span> <span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Recursion O(n)&quot;&quot;&quot;</span><span class="s2">&quot;&quot;</span>
<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Recursion O(n)&quot;&quot;&quot;</span>
<a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a> <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a> <span class="k">return</span>
<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a> <span class="k">return</span> <span class="n">recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
@@ -2858,7 +2864,7 @@
<li>The <code>loop()</code> function calls <code>function()</code> <span class="arithmatex">\(n\)</span> times in a loop, where each iteration's <code>function()</code> returns and releases its stack frame space, so the space complexity remains <span class="arithmatex">\(O(1)\)</span>.</li>
<li>The recursive function <code>recur()</code> will have <span class="arithmatex">\(n\)</span> instances of unreturned <code>recur()</code> existing simultaneously during its execution, thus occupying <span class="arithmatex">\(O(n)\)</span> stack frame space.</li>
</ul>
<h2 id="243-common-types">2.4.3 &nbsp; Common Types<a class="headerlink" href="#243-common-types" title="Permanent link">&para;</a></h2>
<h2 id="243-common-types">2.4.3 &nbsp; Common types<a class="headerlink" href="#243-common-types" title="Permanent link">&para;</a></h2>
<p>Let the size of the input data be <span class="arithmatex">\(n\)</span>, the following chart displays common types of space complexities (arranged from low to high).</p>
<div class="arithmatex">\[
\begin{aligned}
@@ -2866,10 +2872,10 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
\text{Constant Order} &lt; \text{Logarithmic Order} &lt; \text{Linear Order} &lt; \text{Quadratic Order} &lt; \text{Exponential Order}
\end{aligned}
\]</div>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common Types of Space Complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_common_types.png" /></a></p>
<p align="center"> Figure 2-16 &nbsp; Common Types of Space Complexity </p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common types of space complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_common_types.png" /></a></p>
<p align="center"> Figure 2-16 &nbsp; Common types of space complexity </p>
<h3 id="1-constant-order-o1">1. &nbsp; Constant Order <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-order-o1" title="Permanent link">&para;</a></h3>
<h3 id="1-constant-order-o1">1. &nbsp; Constant order <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-order-o1" title="Permanent link">&para;</a></h3>
<p>Constant order is common in constants, variables, objects that are independent of the size of input data <span class="arithmatex">\(n\)</span>.</p>
<p>Note that memory occupied by initializing variables or calling functions in a loop, which is released upon entering the next cycle, does not accumulate over space, thus the space complexity remains <span class="arithmatex">\(O(1)\)</span>:</p>
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@@ -3237,7 +3243,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20function%28%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%87%BD%E6%95%B0%22%22%22%0A%20%20%20%20%23%20%E6%89%A7%E8%A1%8C%E6%9F%90%E4%BA%9B%E6%93%8D%E4%BD%9C%0A%20%20%20%20return%200%0A%0Adef%20constant%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%B8%B8%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E5%B8%B8%E9%87%8F%E3%80%81%E5%8F%98%E9%87%8F%E3%80%81%E5%AF%B9%E8%B1%A1%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20a%20%3D%200%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%2010%0A%20%20%20%20node%20%3D%20ListNode%280%29%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%8F%98%E9%87%8F%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20c%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%87%BD%E6%95%B0%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20function%28%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B8%B8%E6%95%B0%E9%98%B6%0A%20%20%20%20constant%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20function%28%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%87%BD%E6%95%B0%22%22%22%0A%20%20%20%20%23%20%E6%89%A7%E8%A1%8C%E6%9F%90%E4%BA%9B%E6%93%8D%E4%BD%9C%0A%20%20%20%20return%200%0A%0Adef%20constant%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%B8%B8%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E5%B8%B8%E9%87%8F%E3%80%81%E5%8F%98%E9%87%8F%E3%80%81%E5%AF%B9%E8%B1%A1%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20a%20%3D%200%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%2010%0A%20%20%20%20node%20%3D%20ListNode%280%29%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%8F%98%E9%87%8F%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20c%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%87%BD%E6%95%B0%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20function%28%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B8%B8%E6%95%B0%E9%98%B6%0A%20%20%20%20constant%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h3 id="2-linear-order-on">2. &nbsp; Linear Order <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-order-on" title="Permanent link">&para;</a></h3>
<h3 id="2-linear-order-on">2. &nbsp; Linear order <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-order-on" title="Permanent link">&para;</a></h3>
<p>Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to <span class="arithmatex">\(n\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:14"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label><label for="__tabbed_5_14">Zig</label></div>
<div class="tabbed-content">
@@ -3662,10 +3668,10 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><div style="height: 441px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20linear_recur%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E7%BA%BF%E6%80%A7%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20print%28%22%E9%80%92%E5%BD%92%20n%20%3D%22,%20n%29%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20linear_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E7%BA%BF%E6%80%A7%E9%98%B6%0A%20%20%20%20linear_recur%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=25&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear_recur%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E7%BA%BF%E6%80%A7%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20print%28%22%E9%80%92%E5%BD%92%20n%20%3D%22,%20n%29%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20linear_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E7%BA%BF%E6%80%A7%E9%98%B6%0A%20%20%20%20linear_recur%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=25&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_recursive_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursive Function Generating Linear Order Space Complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></a></p>
<p align="center"> Figure 2-17 &nbsp; Recursive Function Generating Linear Order Space Complexity </p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_recursive_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursive function generating linear order space complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></a></p>
<p align="center"> Figure 2-17 &nbsp; Recursive function generating linear order space complexity </p>
<h3 id="3-quadratic-order-on2">3. &nbsp; Quadratic Order <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-order-on2" title="Permanent link">&para;</a></h3>
<h3 id="3-quadratic-order-on2">3. &nbsp; Quadratic order <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-order-on2" title="Permanent link">&para;</a></h3>
<p>Quadratic order is common in matrices and graphs, where the number of elements is quadratic to <span class="arithmatex">\(n\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="7:14"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><input id="__tabbed_7_14" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label><label for="__tabbed_7_14">Zig</label></div>
<div class="tabbed-content">
@@ -4057,10 +4063,10 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20quadratic_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E6%95%B0%E7%BB%84%20nums%20%E9%95%BF%E5%BA%A6%E4%B8%BA%20n,%20n-1,%20...,%202,%201%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%20n%0A%20%20%20%20return%20quadratic_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B9%B3%E6%96%B9%E9%98%B6%0A%20%20%20%20quadratic_recur%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=28&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E6%95%B0%E7%BB%84%20nums%20%E9%95%BF%E5%BA%A6%E4%B8%BA%20n,%20n-1,%20...,%202,%201%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%20n%0A%20%20%20%20return%20quadratic_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B9%B3%E6%96%B9%E9%98%B6%0A%20%20%20%20quadratic_recur%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=28&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_recursive_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursive Function Generating Quadratic Order Space Complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></a></p>
<p align="center"> Figure 2-18 &nbsp; Recursive Function Generating Quadratic Order Space Complexity </p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_recursive_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Recursive function generating quadratic order space complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></a></p>
<p align="center"> Figure 2-18 &nbsp; Recursive function generating quadratic order space complexity </p>
<h3 id="4-exponential-order-o2n">4. &nbsp; Exponential Order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
<h3 id="4-exponential-order-o2n">4. &nbsp; Exponential order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
<p>Exponential order is common in binary trees. Observe the below image, a "full binary tree" with <span class="arithmatex">\(n\)</span> levels has <span class="arithmatex">\(2^n - 1\)</span> nodes, occupying <span class="arithmatex">\(O(2^n)\)</span> space:</p>
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@@ -4238,13 +4244,13 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20build_tree%28n%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BB%BA%E7%AB%8B%E6%BB%A1%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20root%20%3D%20TreeNode%280%29%0A%20%20%20%20root.left%20%3D%20build_tree%28n%20-%201%29%0A%20%20%20%20root.right%20%3D%20build_tree%28n%20-%201%29%0A%20%20%20%20return%20root%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E6%8C%87%E6%95%B0%E9%98%B6%0A%20%20%20%20root%20%3D%20build_tree%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=507&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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<p><a class="glightbox" href="../space_complexity.assets/space_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Full Binary Tree Generating Exponential Order Space Complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_exponential.png" /></a></p>
<p align="center"> Figure 2-19 &nbsp; Full Binary Tree Generating Exponential Order Space Complexity </p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Full binary tree generating exponential order space complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_exponential.png" /></a></p>
<p align="center"> Figure 2-19 &nbsp; Full binary tree generating exponential order space complexity </p>
<h3 id="5-logarithmic-order-olog-n">5. &nbsp; Logarithmic Order <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-order-olog-n" title="Permanent link">&para;</a></h3>
<h3 id="5-logarithmic-order-olog-n">5. &nbsp; Logarithmic order <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-order-olog-n" title="Permanent link">&para;</a></h3>
<p>Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length <span class="arithmatex">\(n\)</span> is recursively divided in half each round, forming a recursion tree of height <span class="arithmatex">\(\log n\)</span>, using <span class="arithmatex">\(O(\log n)\)</span> stack frame space.</p>
<p>Another example is converting a number to a string. Given a positive integer <span class="arithmatex">\(n\)</span>, its number of digits is <span class="arithmatex">\(\log_{10} n + 1\)</span>, corresponding to the length of the string, thus the space complexity is <span class="arithmatex">\(O(\log_{10} n + 1) = O(\log n)\)</span>.</p>
<h2 id="244-balancing-time-and-space">2.4.4 &nbsp; Balancing Time and Space<a class="headerlink" href="#244-balancing-time-and-space" title="Permanent link">&para;</a></h2>
<h2 id="244-balancing-time-and-space">2.4.4 &nbsp; Balancing time and space<a class="headerlink" href="#244-balancing-time-and-space" title="Permanent link">&para;</a></h2>
<p>Ideally, we aim for both time complexity and space complexity to be optimal. However, in practice, optimizing both simultaneously is often difficult.</p>
<p><strong>Lowering time complexity usually comes at the cost of increased space complexity, and vice versa</strong>. The approach of sacrificing memory space to improve algorithm speed is known as "space-time tradeoff"; the reverse is known as "time-space tradeoff".</p>
<p>The choice depends on which aspect we value more. In most cases, time is more precious than space, so "space-time tradeoff" is often the more common strategy. Of course, controlling space complexity is also very important when dealing with large volumes of data.</p>
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
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<p><strong>Algorithm Efficiency Assessment</strong></p>
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<li>Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm.</li>
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<title>2.3 Time Complexity - Hello Algo</title>
<title>2.3 Time complexity - Hello Algo</title>
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2.3 Time Complexity
2.3 Time complexity
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中文
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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2.3.1 &nbsp; Assessing Time Growth Trend
2.3.1 &nbsp; Assessing time growth trend
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2.3.2 &nbsp; Asymptotic Upper Bound
2.3.2 &nbsp; Asymptotic upper bound
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2.3.3 &nbsp; Calculation Method
2.3.3 &nbsp; Calculation method
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1. &nbsp; Step 1: Counting the Number of Operations
1. &nbsp; Step 1: counting the number of operations
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2. &nbsp; Step 2: Determining the Asymptotic Upper Bound
2. &nbsp; Step 2: determining the asymptotic upper bound
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2.3.4 &nbsp; Common Types of Time Complexity
2.3.4 &nbsp; Common types of time complexity
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1. &nbsp; Constant Order
1. &nbsp; Constant order
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2. &nbsp; Linear Order
2. &nbsp; Linear order
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3. &nbsp; Quadratic Order
3. &nbsp; Quadratic order
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<a href="#4-exponential-order-o2n" class="md-nav__link">
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4. &nbsp; Exponential Order
4. &nbsp; Exponential order
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5. &nbsp; Logarithmic Order
5. &nbsp; Logarithmic order
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6. &nbsp; Linear-Logarithmic Order
6. &nbsp; Linear-logarithmic order
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7. &nbsp; Factorial Order
7. &nbsp; Factorial order
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<a href="#235-worst-best-and-average-time-complexities" class="md-nav__link">
<span class="md-ellipsis">
2.3.5 &nbsp; Worst, Best, and Average Time Complexities
2.3.5 &nbsp; Worst, best, and average time complexities
</span>
</a>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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<li class="md-nav__item">
<a href="#231-assessing-time-growth-trend" class="md-nav__link">
<span class="md-ellipsis">
2.3.1 &nbsp; Assessing Time Growth Trend
2.3.1 &nbsp; Assessing time growth trend
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<li class="md-nav__item">
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<span class="md-ellipsis">
2.3.2 &nbsp; Asymptotic Upper Bound
2.3.2 &nbsp; Asymptotic upper bound
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<li class="md-nav__item">
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2.3.3 &nbsp; Calculation Method
2.3.3 &nbsp; Calculation method
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<nav class="md-nav" aria-label="2.3.3   Calculation method">
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<span class="md-ellipsis">
1. &nbsp; Step 1: Counting the Number of Operations
1. &nbsp; Step 1: counting the number of operations
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<span class="md-ellipsis">
2. &nbsp; Step 2: Determining the Asymptotic Upper Bound
2. &nbsp; Step 2: determining the asymptotic upper bound
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<li class="md-nav__item">
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2.3.4 &nbsp; Common Types of Time Complexity
2.3.4 &nbsp; Common types of time complexity
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<nav class="md-nav" aria-label="2.3.4   Common types of time complexity">
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1. &nbsp; Constant Order
1. &nbsp; Constant order
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2. &nbsp; Linear Order
2. &nbsp; Linear order
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3. &nbsp; Quadratic Order
3. &nbsp; Quadratic order
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<a href="#4-exponential-order-o2n" class="md-nav__link">
<span class="md-ellipsis">
4. &nbsp; Exponential Order
4. &nbsp; Exponential order
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<span class="md-ellipsis">
5. &nbsp; Logarithmic Order
5. &nbsp; Logarithmic order
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<li class="md-nav__item">
<a href="#6-linear-logarithmic-order-on-log-n" class="md-nav__link">
<span class="md-ellipsis">
6. &nbsp; Linear-Logarithmic Order
6. &nbsp; Linear-logarithmic order
</span>
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<li class="md-nav__item">
<a href="#7-factorial-order-on" class="md-nav__link">
<span class="md-ellipsis">
7. &nbsp; Factorial Order
7. &nbsp; Factorial order
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<li class="md-nav__item">
<a href="#235-worst-best-and-average-time-complexities" class="md-nav__link">
<span class="md-ellipsis">
2.3.5 &nbsp; Worst, Best, and Average Time Complexities
2.3.5 &nbsp; Worst, best, and average time complexities
</span>
</a>
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<!-- Page content -->
<h1 id="23-time-complexity">2.3 &nbsp; Time Complexity<a class="headerlink" href="#23-time-complexity" title="Permanent link">&para;</a></h1>
<h1 id="23-time-complexity">2.3 &nbsp; Time complexity<a class="headerlink" href="#23-time-complexity" title="Permanent link">&para;</a></h1>
<p>Time complexity is a concept used to measure how the run time of an algorithm increases with the size of the input data. Understanding time complexity is crucial for accurately assessing the efficiency of an algorithm.</p>
<ol>
<li><strong>Determining the Running Platform</strong>: This includes hardware configuration, programming language, system environment, etc., all of which can affect the efficiency of code execution.</li>
@@ -2485,7 +2491,7 @@
1 + 1 + 10 + (1 + 5) \times n = 6n + 12
\]</div>
<p>However, in practice, <strong>counting the run time of an algorithm is neither practical nor reasonable</strong>. First, we don't want to tie the estimated time to the running platform, as algorithms need to run on various platforms. Second, it's challenging to know the run time for each type of operation, making the estimation process difficult.</p>
<h2 id="231-assessing-time-growth-trend">2.3.1 &nbsp; Assessing Time Growth Trend<a class="headerlink" href="#231-assessing-time-growth-trend" title="Permanent link">&para;</a></h2>
<h2 id="231-assessing-time-growth-trend">2.3.1 &nbsp; Assessing time growth trend<a class="headerlink" href="#231-assessing-time-growth-trend" title="Permanent link">&para;</a></h2>
<p>Time complexity analysis does not count the algorithm's run time, <strong>but rather the growth trend of the run time as the data volume increases</strong>.</p>
<p>Let's understand this concept of "time growth trend" with an example. Assume the input data size is <span class="arithmatex">\(n\)</span>, and consider three algorithms <code>A</code>, <code>B</code>, and <code>C</code>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Zig</label></div>
@@ -2729,8 +2735,8 @@
<li>Algorithm <code>B</code> involves a print operation looping <span class="arithmatex">\(n\)</span> times, and its run time grows linearly with <span class="arithmatex">\(n\)</span>. Its time complexity is "linear order."</li>
<li>Algorithm <code>C</code> has a print operation looping 1,000,000 times. Although it takes a long time, it is independent of the input data size <span class="arithmatex">\(n\)</span>. Therefore, the time complexity of <code>C</code> is the same as <code>A</code>, which is "constant order."</li>
</ul>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_simple_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Time Growth Trend of Algorithms A, B, and C" class="animation-figure" src="../time_complexity.assets/time_complexity_simple_example.png" /></a></p>
<p align="center"> Figure 2-7 &nbsp; Time Growth Trend of Algorithms A, B, and C </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_simple_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Time growth trend of algorithms a, b, and c" class="animation-figure" src="../time_complexity.assets/time_complexity_simple_example.png" /></a></p>
<p align="center"> Figure 2-7 &nbsp; Time growth trend of algorithms a, b, and c </p>
<p>Compared to directly counting the run time of an algorithm, what are the characteristics of time complexity analysis?</p>
<ul>
@@ -2738,7 +2744,7 @@
<li><strong>Time complexity analysis is more straightforward</strong>. Obviously, the running platform and the types of computational operations are irrelevant to the trend of run time growth. Therefore, in time complexity analysis, we can simply treat the execution time of all computational operations as the same "unit time," simplifying the "computational operation run time count" to a "computational operation count." This significantly reduces the complexity of estimation.</li>
<li><strong>Time complexity has its limitations</strong>. For example, although algorithms <code>A</code> and <code>C</code> have the same time complexity, their actual run times can be quite different. Similarly, even though algorithm <code>B</code> has a higher time complexity than <code>C</code>, it is clearly superior when the input data size <span class="arithmatex">\(n\)</span> is small. In these cases, it's difficult to judge the efficiency of algorithms based solely on time complexity. Nonetheless, despite these issues, complexity analysis remains the most effective and commonly used method for evaluating algorithm efficiency.</li>
</ul>
<h2 id="232-asymptotic-upper-bound">2.3.2 &nbsp; Asymptotic Upper Bound<a class="headerlink" href="#232-asymptotic-upper-bound" title="Permanent link">&para;</a></h2>
<h2 id="232-asymptotic-upper-bound">2.3.2 &nbsp; Asymptotic upper bound<a class="headerlink" href="#232-asymptotic-upper-bound" title="Permanent link">&para;</a></h2>
<p>Consider a function with an input size of <span class="arithmatex">\(n\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Zig</label></div>
<div class="tabbed-content">
@@ -2902,13 +2908,13 @@ T(n) = 3 + 2n
<p>If there exist positive real numbers <span class="arithmatex">\(c\)</span> and <span class="arithmatex">\(n_0\)</span> such that for all <span class="arithmatex">\(n &gt; n_0\)</span>, <span class="arithmatex">\(T(n) \leq c \cdot f(n)\)</span>, then <span class="arithmatex">\(f(n)\)</span> is considered an asymptotic upper bound of <span class="arithmatex">\(T(n)\)</span>, denoted as <span class="arithmatex">\(T(n) = O(f(n))\)</span>.</p>
</div>
<p>As illustrated below, calculating the asymptotic upper bound involves finding a function <span class="arithmatex">\(f(n)\)</span> such that, as <span class="arithmatex">\(n\)</span> approaches infinity, <span class="arithmatex">\(T(n)\)</span> and <span class="arithmatex">\(f(n)\)</span> have the same growth order, differing only by a constant factor <span class="arithmatex">\(c\)</span>.</p>
<p><a class="glightbox" href="../time_complexity.assets/asymptotic_upper_bound.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Asymptotic Upper Bound of a Function" class="animation-figure" src="../time_complexity.assets/asymptotic_upper_bound.png" /></a></p>
<p align="center"> Figure 2-8 &nbsp; Asymptotic Upper Bound of a Function </p>
<p><a class="glightbox" href="../time_complexity.assets/asymptotic_upper_bound.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Asymptotic upper bound of a function" class="animation-figure" src="../time_complexity.assets/asymptotic_upper_bound.png" /></a></p>
<p align="center"> Figure 2-8 &nbsp; Asymptotic upper bound of a function </p>
<h2 id="233-calculation-method">2.3.3 &nbsp; Calculation Method<a class="headerlink" href="#233-calculation-method" title="Permanent link">&para;</a></h2>
<h2 id="233-calculation-method">2.3.3 &nbsp; Calculation method<a class="headerlink" href="#233-calculation-method" title="Permanent link">&para;</a></h2>
<p>While the concept of asymptotic upper bound might seem mathematically dense, you don't need to fully grasp it right away. Let's first understand the method of calculation, which can be practiced and comprehended over time.</p>
<p>Once <span class="arithmatex">\(f(n)\)</span> is determined, we obtain the time complexity <span class="arithmatex">\(O(f(n))\)</span>. But how do we determine the asymptotic upper bound <span class="arithmatex">\(f(n)\)</span>? This process generally involves two steps: counting the number of operations and determining the asymptotic upper bound.</p>
<h3 id="1-step-1-counting-the-number-of-operations">1. &nbsp; Step 1: Counting the Number of Operations<a class="headerlink" href="#1-step-1-counting-the-number-of-operations" title="Permanent link">&para;</a></h3>
<h3 id="1-step-1-counting-the-number-of-operations">1. &nbsp; Step 1: counting the number of operations<a class="headerlink" href="#1-step-1-counting-the-number-of-operations" title="Permanent link">&para;</a></h3>
<p>This step involves going through the code line by line. However, due to the presence of the constant <span class="arithmatex">\(c\)</span> in <span class="arithmatex">\(c \cdot f(n)\)</span>, <strong>all coefficients and constant terms in <span class="arithmatex">\(T(n)\)</span> can be ignored</strong>. This principle allows for simplification techniques in counting operations.</p>
<ol>
<li><strong>Ignore constant terms in <span class="arithmatex">\(T(n)\)</span></strong>, as they do not affect the time complexity being independent of <span class="arithmatex">\(n\)</span>.</li>
@@ -3136,10 +3142,10 @@ T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \text{Complete Count (-.-|||)} \newl
T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
\end{aligned}
\]</div>
<h3 id="2-step-2-determining-the-asymptotic-upper-bound">2. &nbsp; Step 2: Determining the Asymptotic Upper Bound<a class="headerlink" href="#2-step-2-determining-the-asymptotic-upper-bound" title="Permanent link">&para;</a></h3>
<h3 id="2-step-2-determining-the-asymptotic-upper-bound">2. &nbsp; Step 2: determining the asymptotic upper bound<a class="headerlink" href="#2-step-2-determining-the-asymptotic-upper-bound" title="Permanent link">&para;</a></h3>
<p><strong>The time complexity is determined by the highest order term in <span class="arithmatex">\(T(n)\)</span></strong>. This is because, as <span class="arithmatex">\(n\)</span> approaches infinity, the highest order term dominates, rendering the influence of other terms negligible.</p>
<p>The following table illustrates examples of different operation counts and their corresponding time complexities. Some exaggerated values are used to emphasize that coefficients cannot alter the order of growth. When <span class="arithmatex">\(n\)</span> becomes very large, these constants become insignificant.</p>
<p align="center"> Table: Time Complexity for Different Operation Counts </p>
<p align="center"> Table: Time complexity for different operation counts </p>
<div class="center-table">
<table>
@@ -3173,7 +3179,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</tbody>
</table>
</div>
<h2 id="234-common-types-of-time-complexity">2.3.4 &nbsp; Common Types of Time Complexity<a class="headerlink" href="#234-common-types-of-time-complexity" title="Permanent link">&para;</a></h2>
<h2 id="234-common-types-of-time-complexity">2.3.4 &nbsp; Common types of time complexity<a class="headerlink" href="#234-common-types-of-time-complexity" title="Permanent link">&para;</a></h2>
<p>Let's consider the input data size as <span class="arithmatex">\(n\)</span>. The common types of time complexities are illustrated below, arranged from lowest to highest:</p>
<div class="arithmatex">\[
\begin{aligned}
@@ -3181,10 +3187,10 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
\text{Constant Order} &lt; \text{Logarithmic Order} &lt; \text{Linear Order} &lt; \text{Linear-Logarithmic Order} &lt; \text{Quadratic Order} &lt; \text{Exponential Order} &lt; \text{Factorial Order}
\end{aligned}
\]</div>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common Types of Time Complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_common_types.png" /></a></p>
<p align="center"> Figure 2-9 &nbsp; Common Types of Time Complexity </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common types of time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_common_types.png" /></a></p>
<p align="center"> Figure 2-9 &nbsp; Common types of time complexity </p>
<h3 id="1-constant-order-o1">1. &nbsp; Constant Order <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-order-o1" title="Permanent link">&para;</a></h3>
<h3 id="1-constant-order-o1">1. &nbsp; Constant order <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-order-o1" title="Permanent link">&para;</a></h3>
<p>Constant order means the number of operations is independent of the input data size <span class="arithmatex">\(n\)</span>. In the following function, although the number of operations <code>size</code> might be large, the time complexity remains <span class="arithmatex">\(O(1)\)</span> as it's unrelated to <span class="arithmatex">\(n\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:14"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label><label for="__tabbed_5_14">Zig</label></div>
<div class="tabbed-content">
@@ -3357,7 +3363,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20constant%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B8%B8%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20size%20%3D%2010%0A%20%20%20%20for%20_%20in%20range%28size%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20constant%28n%29%0A%20%20%20%20print%28%22%E5%B8%B8%E6%95%B0%E9%98%B6%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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</details>
<h3 id="2-linear-order-on">2. &nbsp; Linear Order <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-order-on" title="Permanent link">&para;</a></h3>
<h3 id="2-linear-order-on">2. &nbsp; Linear order <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-order-on" title="Permanent link">&para;</a></h3>
<p>Linear order indicates the number of operations grows linearly with the input data size <span class="arithmatex">\(n\)</span>. Linear order commonly appears in single-loop structures:</p>
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<div class="tabbed-content">
@@ -3691,7 +3697,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
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</details>
<p>It's important to note that <strong>the input data size <span class="arithmatex">\(n\)</span> should be determined based on the type of input data</strong>. For example, in the first example, <span class="arithmatex">\(n\)</span> represents the input data size, while in the second example, the length of the array <span class="arithmatex">\(n\)</span> is the data size.</p>
<h3 id="3-quadratic-order-on2">3. &nbsp; Quadratic Order <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-order-on2" title="Permanent link">&para;</a></h3>
<h3 id="3-quadratic-order-on2">3. &nbsp; Quadratic order <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-order-on2" title="Permanent link">&para;</a></h3>
<p>Quadratic order means the number of operations grows quadratically with the input data size <span class="arithmatex">\(n\)</span>. Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of <span class="arithmatex">\(O(n)\)</span>, resulting in an overall complexity of <span class="arithmatex">\(O(n^2)\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="8:14"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><input id="__tabbed_8_13" name="__tabbed_8" type="radio" /><input id="__tabbed_8_14" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Python</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Java</label><label for="__tabbed_8_4">C#</label><label for="__tabbed_8_5">Go</label><label for="__tabbed_8_6">Swift</label><label for="__tabbed_8_7">JS</label><label for="__tabbed_8_8">TS</label><label for="__tabbed_8_9">Dart</label><label for="__tabbed_8_10">Rust</label><label for="__tabbed_8_11">C</label><label for="__tabbed_8_12">Kotlin</label><label for="__tabbed_8_13">Ruby</label><label for="__tabbed_8_14">Zig</label></div>
<div class="tabbed-content">
@@ -3900,8 +3906,8 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E6%AC%A1%E6%95%B0%E4%B8%8E%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%E6%88%90%E5%B9%B3%E6%96%B9%E5%85%B3%E7%B3%BB%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20quadratic%28n%29%0A%20%20%20%20print%28%22%E5%B9%B3%E6%96%B9%E9%98%B6%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>The following image compares constant order, linear order, and quadratic order time complexities.</p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Constant, Linear, and Quadratic Order Time Complexities" class="animation-figure" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></a></p>
<p align="center"> Figure 2-10 &nbsp; Constant, Linear, and Quadratic Order Time Complexities </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Constant, linear, and quadratic order time complexities" class="animation-figure" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></a></p>
<p align="center"> Figure 2-10 &nbsp; Constant, linear, and quadratic order time complexities </p>
<p>For instance, in bubble sort, the outer loop runs <span class="arithmatex">\(n - 1\)</span> times, and the inner loop runs <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(n-2\)</span>, ..., <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span> times, averaging <span class="arithmatex">\(n / 2\)</span> times, resulting in a time complexity of <span class="arithmatex">\(O((n - 1) n / 2) = O(n^2)\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="9:14"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><input id="__tabbed_9_13" name="__tabbed_9" type="radio" /><input id="__tabbed_9_14" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Python</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Java</label><label for="__tabbed_9_4">C#</label><label for="__tabbed_9_5">Go</label><label for="__tabbed_9_6">Swift</label><label for="__tabbed_9_7">JS</label><label for="__tabbed_9_8">TS</label><label for="__tabbed_9_9">Dart</label><label for="__tabbed_9_10">Rust</label><label for="__tabbed_9_11">C</label><label for="__tabbed_9_12">Kotlin</label><label for="__tabbed_9_13">Ruby</label><label for="__tabbed_9_14">Zig</label></div>
@@ -4204,7 +4210,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%EF%BC%88%E5%86%92%E6%B3%A1%E6%8E%92%E5%BA%8F%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%20%20%23%20%E8%AE%A1%E6%95%B0%E5%99%A8%0A%20%20%20%20%23%20%E5%A4%96%E5%BE%AA%E7%8E%AF%EF%BC%9A%E6%9C%AA%E6%8E%92%E5%BA%8F%E5%8C%BA%E9%97%B4%E4%B8%BA%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E5%86%85%E5%BE%AA%E7%8E%AF%EF%BC%9A%E5%B0%86%E6%9C%AA%E6%8E%92%E5%BA%8F%E5%8C%BA%E9%97%B4%20%5B0,%20i%5D%20%E4%B8%AD%E7%9A%84%E6%9C%80%E5%A4%A7%E5%85%83%E7%B4%A0%E4%BA%A4%E6%8D%A2%E8%87%B3%E8%AF%A5%E5%8C%BA%E9%97%B4%E7%9A%84%E6%9C%80%E5%8F%B3%E7%AB%AF%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%BA%A4%E6%8D%A2%20nums%5Bj%5D%20%E4%B8%8E%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20tmp%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D%20%3D%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%203%20%20%23%20%E5%85%83%E7%B4%A0%E4%BA%A4%E6%8D%A2%E5%8C%85%E5%90%AB%203%20%E4%B8%AA%E5%8D%95%E5%85%83%E6%93%8D%E4%BD%9C%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20nums%20%3D%20%5Bi%20for%20i%20in%20range%28n,%200,%20-1%29%5D%20%20%23%20%5Bn,%20n-1,%20...,%202,%201%5D%0A%20%20%20%20count%20%3D%20bubble_sort%28nums%29%0A%20%20%20%20print%28%22%E5%B9%B3%E6%96%B9%E9%98%B6%EF%BC%88%E5%86%92%E6%B3%A1%E6%8E%92%E5%BA%8F%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%EF%BC%88%E5%86%92%E6%B3%A1%E6%8E%92%E5%BA%8F%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%20%20%23%20%E8%AE%A1%E6%95%B0%E5%99%A8%0A%20%20%20%20%23%20%E5%A4%96%E5%BE%AA%E7%8E%AF%EF%BC%9A%E6%9C%AA%E6%8E%92%E5%BA%8F%E5%8C%BA%E9%97%B4%E4%B8%BA%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E5%86%85%E5%BE%AA%E7%8E%AF%EF%BC%9A%E5%B0%86%E6%9C%AA%E6%8E%92%E5%BA%8F%E5%8C%BA%E9%97%B4%20%5B0,%20i%5D%20%E4%B8%AD%E7%9A%84%E6%9C%80%E5%A4%A7%E5%85%83%E7%B4%A0%E4%BA%A4%E6%8D%A2%E8%87%B3%E8%AF%A5%E5%8C%BA%E9%97%B4%E7%9A%84%E6%9C%80%E5%8F%B3%E7%AB%AF%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%BA%A4%E6%8D%A2%20nums%5Bj%5D%20%E4%B8%8E%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20tmp%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D%20%3D%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%203%20%20%23%20%E5%85%83%E7%B4%A0%E4%BA%A4%E6%8D%A2%E5%8C%85%E5%90%AB%203%20%E4%B8%AA%E5%8D%95%E5%85%83%E6%93%8D%E4%BD%9C%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20nums%20%3D%20%5Bi%20for%20i%20in%20range%28n,%200,%20-1%29%5D%20%20%23%20%5Bn,%20n-1,%20...,%202,%201%5D%0A%20%20%20%20count%20%3D%20bubble_sort%28nums%29%0A%20%20%20%20print%28%22%E5%B9%B3%E6%96%B9%E9%98%B6%EF%BC%88%E5%86%92%E6%B3%A1%E6%8E%92%E5%BA%8F%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h3 id="4-exponential-order-o2n">4. &nbsp; Exponential Order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
<h3 id="4-exponential-order-o2n">4. &nbsp; Exponential order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
<p>Biological "cell division" is a classic example of exponential order growth: starting with one cell, it becomes two after one division, four after two divisions, and so on, resulting in <span class="arithmatex">\(2^n\)</span> cells after <span class="arithmatex">\(n\)</span> divisions.</p>
<p>The following image and code simulate the cell division process, with a time complexity of <span class="arithmatex">\(O(2^n)\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="10:14"><input checked="checked" id="__tabbed_10_1" name="__tabbed_10" type="radio" /><input id="__tabbed_10_2" name="__tabbed_10" type="radio" /><input id="__tabbed_10_3" name="__tabbed_10" type="radio" /><input id="__tabbed_10_4" name="__tabbed_10" type="radio" /><input id="__tabbed_10_5" name="__tabbed_10" type="radio" /><input id="__tabbed_10_6" name="__tabbed_10" type="radio" /><input id="__tabbed_10_7" name="__tabbed_10" type="radio" /><input id="__tabbed_10_8" name="__tabbed_10" type="radio" /><input id="__tabbed_10_9" name="__tabbed_10" type="radio" /><input id="__tabbed_10_10" name="__tabbed_10" type="radio" /><input id="__tabbed_10_11" name="__tabbed_10" type="radio" /><input id="__tabbed_10_12" name="__tabbed_10" type="radio" /><input id="__tabbed_10_13" name="__tabbed_10" type="radio" /><input id="__tabbed_10_14" name="__tabbed_10" type="radio" /><div class="tabbed-labels"><label for="__tabbed_10_1">Python</label><label for="__tabbed_10_2">C++</label><label for="__tabbed_10_3">Java</label><label for="__tabbed_10_4">C#</label><label for="__tabbed_10_5">Go</label><label for="__tabbed_10_6">Swift</label><label for="__tabbed_10_7">JS</label><label for="__tabbed_10_8">TS</label><label for="__tabbed_10_9">Dart</label><label for="__tabbed_10_10">Rust</label><label for="__tabbed_10_11">C</label><label for="__tabbed_10_12">Kotlin</label><label for="__tabbed_10_13">Ruby</label><label for="__tabbed_10_14">Zig</label></div>
@@ -4447,8 +4453,8 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<p><div style="height: 531px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20exponential%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20base%20%3D%201%0A%20%20%20%20%23%20%E7%BB%86%E8%83%9E%E6%AF%8F%E8%BD%AE%E4%B8%80%E5%88%86%E4%B8%BA%E4%BA%8C%EF%BC%8C%E5%BD%A2%E6%88%90%E6%95%B0%E5%88%97%201,%202,%204,%208,%20...,%202%5E%28n-1%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28base%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%20%20%20%20base%20*%3D%202%0A%20%20%20%20%23%20count%20%3D%201%20%2B%202%20%2B%204%20%2B%208%20%2B%20..%20%2B%202%5E%28n-1%29%20%3D%202%5En%20-%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exponential%28n%29%0A%20%20%20%20print%28%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20exponential%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20base%20%3D%201%0A%20%20%20%20%23%20%E7%BB%86%E8%83%9E%E6%AF%8F%E8%BD%AE%E4%B8%80%E5%88%86%E4%B8%BA%E4%BA%8C%EF%BC%8C%E5%BD%A2%E6%88%90%E6%95%B0%E5%88%97%201,%202,%204,%208,%20...,%202%5E%28n-1%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28base%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%20%20%20%20base%20*%3D%202%0A%20%20%20%20%23%20count%20%3D%201%20%2B%202%20%2B%204%20%2B%208%20%2B%20..%20%2B%202%5E%28n-1%29%20%3D%202%5En%20-%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exponential%28n%29%0A%20%20%20%20print%28%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Exponential Order Time Complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_exponential.png" /></a></p>
<p align="center"> Figure 2-11 &nbsp; Exponential Order Time Complexity </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Exponential order time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_exponential.png" /></a></p>
<p align="center"> Figure 2-11 &nbsp; Exponential order time complexity </p>
<p>In practice, exponential order often appears in recursive functions. For example, in the code below, it recursively splits into two halves, stopping after <span class="arithmatex">\(n\)</span> divisions:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="11:14"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><input id="__tabbed_11_9" name="__tabbed_11" type="radio" /><input id="__tabbed_11_10" name="__tabbed_11" type="radio" /><input id="__tabbed_11_11" name="__tabbed_11" type="radio" /><input id="__tabbed_11_12" name="__tabbed_11" type="radio" /><input id="__tabbed_11_13" name="__tabbed_11" type="radio" /><input id="__tabbed_11_14" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Python</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Java</label><label for="__tabbed_11_4">C#</label><label for="__tabbed_11_5">Go</label><label for="__tabbed_11_6">Swift</label><label for="__tabbed_11_7">JS</label><label for="__tabbed_11_8">TS</label><label for="__tabbed_11_9">Dart</label><label for="__tabbed_11_10">Rust</label><label for="__tabbed_11_11">C</label><label for="__tabbed_11_12">Kotlin</label><label for="__tabbed_11_13">Ruby</label><label for="__tabbed_11_14">Zig</label></div>
@@ -4584,7 +4590,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20exp_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20return%20exp_recur%28n%20-%201%29%20%2B%20exp_recur%28n%20-%201%29%20%2B%201%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%207%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exp_recur%28n%29%0A%20%20%20%20print%28%22%E6%8C%87%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>Exponential order growth is extremely rapid and is commonly seen in exhaustive search methods (brute force, backtracking, etc.). For large-scale problems, exponential order is unacceptable, often requiring dynamic programming or greedy algorithms as solutions.</p>
<h3 id="5-logarithmic-order-olog-n">5. &nbsp; Logarithmic Order <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-order-olog-n" title="Permanent link">&para;</a></h3>
<h3 id="5-logarithmic-order-olog-n">5. &nbsp; Logarithmic order <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-order-olog-n" title="Permanent link">&para;</a></h3>
<p>In contrast to exponential order, logarithmic order reflects situations where "the size is halved each round." Given an input data size <span class="arithmatex">\(n\)</span>, since the size is halved each round, the number of iterations is <span class="arithmatex">\(\log_2 n\)</span>, the inverse function of <span class="arithmatex">\(2^n\)</span>.</p>
<p>The following image and code simulate the "halving each round" process, with a time complexity of <span class="arithmatex">\(O(\log_2 n)\)</span>, commonly abbreviated as <span class="arithmatex">\(O(\log n)\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="12:14"><input checked="checked" id="__tabbed_12_1" name="__tabbed_12" type="radio" /><input id="__tabbed_12_2" name="__tabbed_12" type="radio" /><input id="__tabbed_12_3" name="__tabbed_12" type="radio" /><input id="__tabbed_12_4" name="__tabbed_12" type="radio" /><input id="__tabbed_12_5" name="__tabbed_12" type="radio" /><input id="__tabbed_12_6" name="__tabbed_12" type="radio" /><input id="__tabbed_12_7" name="__tabbed_12" type="radio" /><input id="__tabbed_12_8" name="__tabbed_12" type="radio" /><input id="__tabbed_12_9" name="__tabbed_12" type="radio" /><input id="__tabbed_12_10" name="__tabbed_12" type="radio" /><input id="__tabbed_12_11" name="__tabbed_12" type="radio" /><input id="__tabbed_12_12" name="__tabbed_12" type="radio" /><input id="__tabbed_12_13" name="__tabbed_12" type="radio" /><input id="__tabbed_12_14" name="__tabbed_12" type="radio" /><div class="tabbed-labels"><label for="__tabbed_12_1">Python</label><label for="__tabbed_12_2">C++</label><label for="__tabbed_12_3">Java</label><label for="__tabbed_12_4">C#</label><label for="__tabbed_12_5">Go</label><label for="__tabbed_12_6">Swift</label><label for="__tabbed_12_7">JS</label><label for="__tabbed_12_8">TS</label><label for="__tabbed_12_9">Dart</label><label for="__tabbed_12_10">Rust</label><label for="__tabbed_12_11">C</label><label for="__tabbed_12_12">Kotlin</label><label for="__tabbed_12_13">Ruby</label><label for="__tabbed_12_14">Zig</label></div>
@@ -4768,8 +4774,8 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E5%BE%AA%E7%8E%AF%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Logarithmic Order Time Complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic.png" /></a></p>
<p align="center"> Figure 2-12 &nbsp; Logarithmic Order Time Complexity </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Logarithmic order time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic.png" /></a></p>
<p align="center"> Figure 2-12 &nbsp; Logarithmic order time complexity </p>
<p>Like exponential order, logarithmic order also frequently appears in recursive functions. The code below forms a recursive tree of height <span class="arithmatex">\(\log_2 n\)</span>:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="13:14"><input checked="checked" id="__tabbed_13_1" name="__tabbed_13" type="radio" /><input id="__tabbed_13_2" name="__tabbed_13" type="radio" /><input id="__tabbed_13_3" name="__tabbed_13" type="radio" /><input id="__tabbed_13_4" name="__tabbed_13" type="radio" /><input id="__tabbed_13_5" name="__tabbed_13" type="radio" /><input id="__tabbed_13_6" name="__tabbed_13" type="radio" /><input id="__tabbed_13_7" name="__tabbed_13" type="radio" /><input id="__tabbed_13_8" name="__tabbed_13" type="radio" /><input id="__tabbed_13_9" name="__tabbed_13" type="radio" /><input id="__tabbed_13_10" name="__tabbed_13" type="radio" /><input id="__tabbed_13_11" name="__tabbed_13" type="radio" /><input id="__tabbed_13_12" name="__tabbed_13" type="radio" /><input id="__tabbed_13_13" name="__tabbed_13" type="radio" /><input id="__tabbed_13_14" name="__tabbed_13" type="radio" /><div class="tabbed-labels"><label for="__tabbed_13_1">Python</label><label for="__tabbed_13_2">C++</label><label for="__tabbed_13_3">Java</label><label for="__tabbed_13_4">C#</label><label for="__tabbed_13_5">Go</label><label for="__tabbed_13_6">Swift</label><label for="__tabbed_13_7">JS</label><label for="__tabbed_13_8">TS</label><label for="__tabbed_13_9">Dart</label><label for="__tabbed_13_10">Rust</label><label for="__tabbed_13_11">C</label><label for="__tabbed_13_12">Kotlin</label><label for="__tabbed_13_13">Ruby</label><label for="__tabbed_13_14">Zig</label></div>
@@ -4912,7 +4918,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
\]</div>
<p>This means the base <span class="arithmatex">\(m\)</span> can be changed without affecting the complexity. Therefore, we often omit the base <span class="arithmatex">\(m\)</span> and simply denote logarithmic order as <span class="arithmatex">\(O(\log n)\)</span>.</p>
</div>
<h3 id="6-linear-logarithmic-order-on-log-n">6. &nbsp; Linear-Logarithmic Order <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#6-linear-logarithmic-order-on-log-n" title="Permanent link">&para;</a></h3>
<h3 id="6-linear-logarithmic-order-on-log-n">6. &nbsp; Linear-logarithmic order <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#6-linear-logarithmic-order-on-log-n" title="Permanent link">&para;</a></h3>
<p>Linear-logarithmic order often appears in nested loops, with the complexities of the two loops being <span class="arithmatex">\(O(\log n)\)</span> and <span class="arithmatex">\(O(n)\)</span> respectively. The related code is as follows:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="14:14"><input checked="checked" id="__tabbed_14_1" name="__tabbed_14" type="radio" /><input id="__tabbed_14_2" name="__tabbed_14" type="radio" /><input id="__tabbed_14_3" name="__tabbed_14" type="radio" /><input id="__tabbed_14_4" name="__tabbed_14" type="radio" /><input id="__tabbed_14_5" name="__tabbed_14" type="radio" /><input id="__tabbed_14_6" name="__tabbed_14" type="radio" /><input id="__tabbed_14_7" name="__tabbed_14" type="radio" /><input id="__tabbed_14_8" name="__tabbed_14" type="radio" /><input id="__tabbed_14_9" name="__tabbed_14" type="radio" /><input id="__tabbed_14_10" name="__tabbed_14" type="radio" /><input id="__tabbed_14_11" name="__tabbed_14" type="radio" /><input id="__tabbed_14_12" name="__tabbed_14" type="radio" /><input id="__tabbed_14_13" name="__tabbed_14" type="radio" /><input id="__tabbed_14_14" name="__tabbed_14" type="radio" /><div class="tabbed-labels"><label for="__tabbed_14_1">Python</label><label for="__tabbed_14_2">C++</label><label for="__tabbed_14_3">Java</label><label for="__tabbed_14_4">C#</label><label for="__tabbed_14_5">Go</label><label for="__tabbed_14_6">Swift</label><label for="__tabbed_14_7">JS</label><label for="__tabbed_14_8">TS</label><label for="__tabbed_14_9">Dart</label><label for="__tabbed_14_10">Rust</label><label for="__tabbed_14_11">C</label><label for="__tabbed_14_12">Kotlin</label><label for="__tabbed_14_13">Ruby</label><label for="__tabbed_14_14">Zig</label></div>
<div class="tabbed-content">
@@ -5102,11 +5108,11 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear_log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%BA%BF%E6%80%A7%E5%AF%B9%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%20//%202%29%20%2B%20linear_log_recur%28n%20//%202%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%29%0A%20%20%20%20print%28%22%E7%BA%BF%E6%80%A7%E5%AF%B9%E6%95%B0%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>The image below demonstrates how linear-logarithmic order is generated. Each level of a binary tree has <span class="arithmatex">\(n\)</span> operations, and the tree has <span class="arithmatex">\(\log_2 n + 1\)</span> levels, resulting in a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>.</p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear-Logarithmic Order Time Complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></a></p>
<p align="center"> Figure 2-13 &nbsp; Linear-Logarithmic Order Time Complexity </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear-logarithmic order time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></a></p>
<p align="center"> Figure 2-13 &nbsp; Linear-logarithmic order time complexity </p>
<p>Mainstream sorting algorithms typically have a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>, such as quicksort, mergesort, and heapsort.</p>
<h3 id="7-factorial-order-on">7. &nbsp; Factorial Order <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#7-factorial-order-on" title="Permanent link">&para;</a></h3>
<h3 id="7-factorial-order-on">7. &nbsp; Factorial order <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#7-factorial-order-on" title="Permanent link">&para;</a></h3>
<p>Factorial order corresponds to the mathematical problem of "full permutation." Given <span class="arithmatex">\(n\)</span> distinct elements, the total number of possible permutations is:</p>
<div class="arithmatex">\[
n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20factorial_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%98%B6%E4%B9%98%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20%E4%BB%8E%201%20%E4%B8%AA%E5%88%86%E8%A3%82%E5%87%BA%20n%20%E4%B8%AA%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20factorial_recur%28n%20-%201%29%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20factorial_recur%28n%29%0A%20%20%20%20print%28%22%E9%98%B6%E4%B9%98%E9%98%B6%EF%BC%88%E9%80%92%E5%BD%92%E5%AE%9E%E7%8E%B0%EF%BC%89%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_factorial.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Factorial Order Time Complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_factorial.png" /></a></p>
<p align="center"> Figure 2-14 &nbsp; Factorial Order Time Complexity </p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_factorial.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Factorial order time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_factorial.png" /></a></p>
<p align="center"> Figure 2-14 &nbsp; Factorial order time complexity </p>
<p>Note that factorial order grows even faster than exponential order; it's unacceptable for larger <span class="arithmatex">\(n\)</span> values.</p>
<h2 id="235-worst-best-and-average-time-complexities">2.3.5 &nbsp; Worst, Best, and Average Time Complexities<a class="headerlink" href="#235-worst-best-and-average-time-complexities" title="Permanent link">&para;</a></h2>
<h2 id="235-worst-best-and-average-time-complexities">2.3.5 &nbsp; Worst, best, and average time complexities<a class="headerlink" href="#235-worst-best-and-average-time-complexities" title="Permanent link">&para;</a></h2>
<p><strong>The time efficiency of an algorithm is often not fixed but depends on the distribution of the input data</strong>. Assume we have an array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, consisting of numbers from <span class="arithmatex">\(1\)</span> to <span class="arithmatex">\(n\)</span>, each appearing only once, but in a randomly shuffled order. The task is to return the index of the element <span class="arithmatex">\(1\)</span>. We can draw the following conclusions:</p>
<ul>
<li>When <code>nums = [?, ?, ..., 1]</code>, that is, when the last element is <span class="arithmatex">\(1\)</span>, it requires a complete traversal of the array, <strong>achieving the worst-case time complexity of <span class="arithmatex">\(O(n)\)</span></strong>.</li>
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<title>3.2 Fundamental Data Types - Hello Algo</title>
<title>3.2 Fundamental data types - Hello Algo</title>
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3.2 Fundamental Data Types
3.2 Fundamental data types
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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繁體中文
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
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Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<!-- Page content -->
<h1 id="32-basic-data-types">3.2 &nbsp; Basic Data Types<a class="headerlink" href="#32-basic-data-types" title="Permanent link">&para;</a></h1>
<h1 id="32-basic-data-types">3.2 &nbsp; Basic data types<a class="headerlink" href="#32-basic-data-types" title="Permanent link">&para;</a></h1>
<p>When discussing data in computers, various forms like text, images, videos, voice and 3D models comes to mind. Despite their different organizational forms, they are all composed of various basic data types.</p>
<p><strong>Basic data types are those that the CPU can directly operate on</strong> and are directly used in algorithms, mainly including the following.</p>
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<li>The integer type <code>int</code> occupies 4 bytes = 32 bits and can represent <span class="arithmatex">\(2^{32}\)</span> numbers.</li>
</ul>
<p>The following table lists the space occupied, value range, and default values of various basic data types in Java. While memorizing this table isn't necessary, having a general understanding of it and referencing it when required is recommended.</p>
<p align="center"> Table 3-1 &nbsp; Space Occupied and Value Range of Basic Data Types </p>
<p align="center"> Table 3-1 &nbsp; Space occupied and value range of basic data types </p>
<div class="center-table">
<table>
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<title>3.4 Character Encoding * - Hello Algo</title>
<title>3.4 Character encoding * - Hello Algo</title>
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3.4 Character Encoding *
3.4 Character encoding *
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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繁體中文
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character encoding *
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3.4 Character encoding *
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3.4.1 &nbsp; ASCII character set
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3.4.2 &nbsp; GBK character set
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3.4.3 &nbsp; Unicode Character Set
3.4.3 &nbsp; Unicode character set
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3.4.4 &nbsp; UTF-8 encoding
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3.4.5 &nbsp; Character Encoding in Programming Languages
3.4.5 &nbsp; Character encoding in programming languages
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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3.4.2 &nbsp; GBK character set
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3.4.3 &nbsp; Unicode character set
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3.4.4 &nbsp; UTF-8 encoding
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3.4.5 &nbsp; Character encoding in programming languages
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<!-- Page content -->
<h1 id="34-character-encoding">3.4 &nbsp; Character Encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
<h1 id="34-character-encoding">3.4 &nbsp; Character encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
<p>In the computer system, all data is stored in binary form, and characters (represented by char) are no exception. To represent characters, we need to develop a "character set" that defines a one-to-one mapping between each character and binary numbers. With the character set, computers can convert binary numbers to characters by looking up the table.</p>
<h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII Character Set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
<h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII character set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
<p>The "ASCII code" is one of the earliest character sets, officially known as the American Standard Code for Information Interchange. It uses 7 binary digits (the lower 7 bits of a byte) to represent a character, allowing for a maximum of 128 different characters. As shown in the Figure 3-6 , ASCII includes uppercase and lowercase English letters, numbers 0 ~ 9, various punctuation marks, and certain control characters (such as newline and tab).</p>
<p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII Code" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
<p align="center"> Figure 3-6 &nbsp; ASCII Code </p>
<p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII code" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
<p align="center"> Figure 3-6 &nbsp; ASCII code </p>
<p>However, <strong>ASCII can only represent English characters</strong>. With the globalization of computers, a character set called "EASCII" was developed to represent more languages. It expands from the 7-bit structure of ASCII to 8 bits, enabling the representation of 256 characters.</p>
<p>Globally, various region-specific EASCII character sets have been introduced. The first 128 characters of these sets are consistent with the ASCII, while the remaining 128 characters are defined differently to accommodate the requirements of different languages.</p>
<h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK Character Set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
<h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK character set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
<p>Later, it was found that <strong>EASCII still could not meet the character requirements of many languages</strong>. For instance, there are nearly a hundred thousand Chinese characters, with several thousand used regularly. In 1980, the Standardization Administration of China released the "GB2312" character set, which included 6763 Chinese characters, essentially fulfilling the computer processing needs for the Chinese language.</p>
<p>However, GB2312 could not handle some rare and traditional characters. The "GBK" character set expands GB2312 and includes 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented with one byte, while Chinese characters use two bytes.</p>
<h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode Character Set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
<h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode character set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
<p>With the rapid evolution of computer technology and a plethora of character sets and encoding standards, numerous problems arose. On the one hand, these character sets generally only defined characters for specific languages and could not function properly in multilingual environments. On the other hand, the existence of multiple character set standards for the same language caused garbled text when information was exchanged between computers using different encoding standards.</p>
<p>Researchers of that era thought: <strong>What if a comprehensive character set encompassing all global languages and symbols was developed? Wouldn't this resolve the issues associated with cross-linguistic environments and garbled text?</strong> Inspired by this idea, the extensive character set, Unicode, was born.</p>
<p>"Unicode" is referred to as "统一码" (Unified Code) in Chinese, theoretically capable of accommodating over a million characters. It aims to incorporate characters from all over the world into a single set, providing a universal character set for processing and displaying various languages and reducing the issues of garbled text due to different encoding standards.</p>
<p>Since its release in 1991, Unicode has continually expanded to include new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emojis from various languages. In the vast Unicode character set, commonly used characters occupy 2 bytes, while some rare characters may occupy 3 or even 4 bytes.</p>
<p>Unicode is a universal character set that assigns a number (called a "code point") to each character, <strong>but it does not specify how these character code points should be stored in a computer system</strong>. One might ask: How does a system interpret Unicode code points of varying lengths within a text? For example, given a 2-byte code, how does the system determine if it represents a single 2-byte character or two 1-byte characters?</p>
<p>A straightforward solution to this problem is to store all characters as equal-length encodings. As shown in the Figure 3-7 , each character in "Hello" occupies 1 byte, while each character in "算法" (algorithm) occupies 2 bytes. We could encode all characters in "Hello 算法" as 2 bytes by padding the higher bits with zeros. This method would enable the system to interpret a character every 2 bytes, recovering the content of the phrase.</p>
<p><a class="glightbox" href="../character_encoding.assets/unicode_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Unicode Encoding Example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></a></p>
<p align="center"> Figure 3-7 &nbsp; Unicode Encoding Example </p>
<p><a class="glightbox" href="../character_encoding.assets/unicode_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Unicode encoding example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></a></p>
<p align="center"> Figure 3-7 &nbsp; Unicode encoding example </p>
<p>However, as ASCII has shown us, encoding English only requires 1 byte. Using the above approach would double the space occupied by English text compared to ASCII encoding, which is a waste of memory space. Therefore, a more efficient Unicode encoding method is needed.</p>
<h2 id="344-utf-8-encoding">3.4.4 &nbsp; UTF-8 Encoding<a class="headerlink" href="#344-utf-8-encoding" title="Permanent link">&para;</a></h2>
<h2 id="344-utf-8-encoding">3.4.4 &nbsp; UTF-8 encoding<a class="headerlink" href="#344-utf-8-encoding" title="Permanent link">&para;</a></h2>
<p>Currently, UTF-8 has become the most widely used Unicode encoding method internationally. <strong>It is a variable-length encoding</strong>, using 1 to 4 bytes to represent a character, depending on the complexity of the character. ASCII characters need only 1 byte, Latin and Greek letters require 2 bytes, commonly used Chinese characters need 3 bytes, and some other rare characters need 4 bytes.</p>
<p>The encoding rules for UTF-8 are not complex and can be divided into two cases:</p>
<ul>
@@ -2157,22 +2163,22 @@
<p>The Figure 3-8 shows the UTF-8 encoding for "Hello算法". It can be observed that since the highest <span class="arithmatex">\(n\)</span> bits are set to <span class="arithmatex">\(1\)</span>, the system can determine the length of the character as <span class="arithmatex">\(n\)</span> by counting the number of highest bits set to <span class="arithmatex">\(1\)</span>.</p>
<p>But why set the highest 2 bits of the remaining bytes to <span class="arithmatex">\(10\)</span>? Actually, this <span class="arithmatex">\(10\)</span> serves as a kind of checksum. If the system starts parsing text from an incorrect byte, the <span class="arithmatex">\(10\)</span> at the beginning of the byte can help the system quickly detect anomalies.</p>
<p>The reason for using <span class="arithmatex">\(10\)</span> as a checksum is that, under UTF-8 encoding rules, it's impossible for the highest two bits of a character to be <span class="arithmatex">\(10\)</span>. This can be proven by contradiction: If the highest two bits of a character are <span class="arithmatex">\(10\)</span>, it indicates that the character's length is <span class="arithmatex">\(1\)</span>, corresponding to ASCII. However, the highest bit of an ASCII character should be <span class="arithmatex">\(0\)</span>, which contradicts the assumption.</p>
<p><a class="glightbox" href="../character_encoding.assets/utf-8_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="UTF-8 Encoding Example" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></a></p>
<p align="center"> Figure 3-8 &nbsp; UTF-8 Encoding Example </p>
<p><a class="glightbox" href="../character_encoding.assets/utf-8_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="UTF-8 encoding example" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></a></p>
<p align="center"> Figure 3-8 &nbsp; UTF-8 encoding example </p>
<p>Apart from UTF-8, other common encoding methods include:</p>
<ul>
<li><strong>UTF-16 Encoding</strong>: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters require 4 bytes. For 2-byte characters, the UTF-16 encoding equals the Unicode code point.</li>
<li><strong>UTF-32 Encoding</strong>: Every character uses 4 bytes. This means UTF-32 occupies more space than UTF-8 and UTF-16, especially for texts with a high proportion of ASCII characters.</li>
<li><strong>UTF-16 encoding</strong>: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters require 4 bytes. For 2-byte characters, the UTF-16 encoding equals the Unicode code point.</li>
<li><strong>UTF-32 encoding</strong>: Every character uses 4 bytes. This means UTF-32 occupies more space than UTF-8 and UTF-16, especially for texts with a high proportion of ASCII characters.</li>
</ul>
<p>From the perspective of storage space, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 to encode some non-English characters (such as Chinese) can be more efficient because it only requires 2 bytes, while UTF-8 might need 3 bytes.</p>
<p>From a compatibility perspective, UTF-8 is the most versatile, with many tools and libraries supporting UTF-8 as a priority.</p>
<h2 id="345-character-encoding-in-programming-languages">3.4.5 &nbsp; Character Encoding in Programming Languages<a class="headerlink" href="#345-character-encoding-in-programming-languages" title="Permanent link">&para;</a></h2>
<h2 id="345-character-encoding-in-programming-languages">3.4.5 &nbsp; Character encoding in programming languages<a class="headerlink" href="#345-character-encoding-in-programming-languages" title="Permanent link">&para;</a></h2>
<p>Historically, many programming languages utilized fixed-length encodings such as UTF-16 or UTF-32 for processing strings during program execution. This allows strings to be handled as arrays, offering several advantages:</p>
<ul>
<li><strong>Random Access</strong>: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the <span class="arithmatex">\(i^{th}\)</span> character requires traversing the string from the start to the <span class="arithmatex">\(i^{th}\)</span> position, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
<li><strong>Character Counting</strong>: Similar to random access, counting the number of characters in a UTF-16 encoded string is an <span class="arithmatex">\(O(1)\)</span> operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.</li>
<li><strong>String Operations</strong>: Many string operations like splitting, concatenating, inserting, and deleting are easier on UTF-16 encoded strings. These operations generally require additional computation on UTF-8 encoded strings to ensure the validity of the UTF-8 encoding.</li>
<li><strong>Random access</strong>: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the <span class="arithmatex">\(i^{th}\)</span> character requires traversing the string from the start to the <span class="arithmatex">\(i^{th}\)</span> position, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
<li><strong>Character counting</strong>: Similar to random access, counting the number of characters in a UTF-16 encoded string is an <span class="arithmatex">\(O(1)\)</span> operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.</li>
<li><strong>String operations</strong>: Many string operations like splitting, concatenating, inserting, and deleting are easier on UTF-16 encoded strings. These operations generally require additional computation on UTF-8 encoded strings to ensure the validity of the UTF-8 encoding.</li>
</ul>
<p>The design of character encoding schemes in programming languages is an interesting topic involving various factors:</p>
<ul>
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<title>3.1 Classification of Data Structures - Hello Algo</title>
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3.1 Classification of Data Structures
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.1.1 &nbsp; Logical Structure: Linear and Non-Linear
3.1.1 &nbsp; Logical structure: linear and non-linear
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3.1.2 &nbsp; Physical structure: contiguous and dispersed
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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Chapter 4. Array and Linked List
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Chapter 5. Stack and queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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3.1.1 &nbsp; Logical structure: linear and non-linear
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<!-- Page content -->
<h1 id="31-classification-of-data-structures">3.1 &nbsp; Classification of Data Structures<a class="headerlink" href="#31-classification-of-data-structures" title="Permanent link">&para;</a></h1>
<h1 id="31-classification-of-data-structures">3.1 &nbsp; Classification of data structures<a class="headerlink" href="#31-classification-of-data-structures" title="Permanent link">&para;</a></h1>
<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified into "logical structure" and "physical structure".</p>
<h2 id="311-logical-structure-linear-and-non-linear">3.1.1 &nbsp; Logical Structure: Linear and Non-Linear<a class="headerlink" href="#311-logical-structure-linear-and-non-linear" title="Permanent link">&para;</a></h2>
<h2 id="311-logical-structure-linear-and-non-linear">3.1.1 &nbsp; Logical structure: linear and non-linear<a class="headerlink" href="#311-logical-structure-linear-and-non-linear" title="Permanent link">&para;</a></h2>
<p><strong>The logical structures reveal the logical relationships between data elements</strong>. In arrays and linked lists, data are arranged in a specific sequence, demonstrating the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between "ancestors" and "descendants"; and graphs are composed of nodes and edges, reflecting the intricate network relationship.</p>
<p>As shown in the Figure 3-1 , logical structures can be divided into two major categories: "linear" and "non-linear". Linear structures are more intuitive, indicating data is arranged linearly in logical relationships; non-linear structures, conversely, are arranged non-linearly.</p>
<ul>
<li><strong>Linear Data Structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li>
<li><strong>Non-Linear Data Structures</strong>: Trees, Heaps, Graphs, Hash Tables.</li>
<li><strong>Linear data structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li>
<li><strong>Non-linear data structures</strong>: Trees, Heaps, Graphs, Hash Tables.</li>
</ul>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear and Non-Linear Data Structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
<p align="center"> Figure 3-1 &nbsp; Linear and Non-Linear Data Structures </p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear and non-linear data structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
<p align="center"> Figure 3-1 &nbsp; Linear and non-linear data structures </p>
<p>Non-linear data structures can be further divided into tree structures and network structures.</p>
<ul>
<li><strong>Linear Structures</strong>: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li>
<li><strong>Tree Structures</strong>: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li>
<li><strong>Network Structures</strong>: Graphs, where elements have a many-to-many relationships.</li>
<li><strong>Linear structures</strong>: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li>
<li><strong>Tree structures</strong>: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li>
<li><strong>Network structures</strong>: Graphs, where elements have a many-to-many relationships.</li>
</ul>
<h2 id="312-physical-structure-contiguous-and-dispersed">3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed<a class="headerlink" href="#312-physical-structure-contiguous-and-dispersed" title="Permanent link">&para;</a></h2>
<h2 id="312-physical-structure-contiguous-and-dispersed">3.1.2 &nbsp; Physical structure: contiguous and dispersed<a class="headerlink" href="#312-physical-structure-contiguous-and-dispersed" title="Permanent link">&para;</a></h2>
<p><strong>During the execution of an algorithm, the data being processed is stored in memory</strong>. The Figure 3-2 shows a computer memory stick where each black square is a physical memory space. We can think of memory as a vast Excel spreadsheet, with each cell capable of storing a certain amount of data.</p>
<p><strong>The system accesses the data at the target location by means of a memory address</strong>. As shown in the Figure 3-2 , the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data stored in memory.</p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/computer_memory_location.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory Stick, Memory Spaces, Memory Addresses" class="animation-figure" src="../classification_of_data_structure.assets/computer_memory_location.png" /></a></p>
<p align="center"> Figure 3-2 &nbsp; Memory Stick, Memory Spaces, Memory Addresses </p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/computer_memory_location.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory stick, memory spaces, memory addresses" class="animation-figure" src="../classification_of_data_structure.assets/computer_memory_location.png" /></a></p>
<p align="center"> Figure 3-2 &nbsp; Memory stick, memory spaces, memory addresses </p>
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@@ -2100,8 +2106,8 @@
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<p>Memory is a shared resource for all programs. When a block of memory is occupied by one program, it cannot be simultaneously used by other programs. <strong>Therefore, considering memory resources is crucial in designing data structures and algorithms</strong>. For instance, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.</p>
<p>As illustrated in the Figure 3-3 , <strong>the physical structure reflects the way data is stored in computer memory</strong> and it can be divided into contiguous space storage (arrays) and non-contiguous space storage (linked lists). The two types of physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_phisical_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Contiguous Space Storage and Dispersed Space Storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></a></p>
<p align="center"> Figure 3-3 &nbsp; Contiguous Space Storage and Dispersed Space Storage </p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_phisical_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Contiguous space storage and dispersed space storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></a></p>
<p align="center"> Figure 3-3 &nbsp; Contiguous space storage and dispersed space storage </p>
<p><strong>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both</strong>. For example, stacks and queues can be implemented using either arrays or linked lists; while implementations of hash tables may involve both arrays and linked lists.
- <strong>Array-based implementations</strong>: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions <span class="arithmatex">\(\geq 3\)</span>).
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<title>Chapter 3.   Data Structures - Hello Algo</title>
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Chapter 3. &nbsp; Data Structures
Chapter 3. &nbsp; Data structures
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@@ -201,7 +201,13 @@
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@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
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@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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@@ -642,7 +648,7 @@
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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@@ -998,7 +1004,7 @@
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Chapter 4. Array and Linked List
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@@ -1036,7 +1042,7 @@
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4.2 Linked List
4.2 Linked list
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1157,7 +1163,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1173,7 +1179,7 @@
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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@@ -1232,7 +1238,7 @@
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1311,7 +1317,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1327,7 +1333,7 @@
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Chapter 6. Hash Table
Chapter 6. Hash table
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@@ -1344,7 +1350,7 @@
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6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2003,8 +2009,8 @@
<!-- Page content -->
<h1 id="chapter-3-data-structures">Chapter 3. &nbsp; Data Structures<a class="headerlink" href="#chapter-3-data-structures" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_data_structure.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Data Structures" class="cover-image" src="../assets/covers/chapter_data_structure.jpg" /></a></p>
<h1 id="chapter-3-data-structures">Chapter 3. &nbsp; Data structures<a class="headerlink" href="#chapter-3-data-structures" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_data_structure.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Data structures" class="cover-image" src="../assets/covers/chapter_data_structure.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Data structures serve as a robust and diverse framework.</p>
@@ -2012,10 +2018,10 @@
</div>
<h2 id="chapter-contents">Chapter Contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/classification_of_data_structure/">3.1 &nbsp; Classification of Data Structures</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/basic_data_types/">3.2 &nbsp; Fundamental Data Types</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/number_encoding/">3.3 &nbsp; Number Encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/character_encoding/">3.4 &nbsp; Character Encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/classification_of_data_structure/">3.1 &nbsp; Classification of data structures</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/basic_data_types/">3.2 &nbsp; Fundamental data types</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/number_encoding/">3.3 &nbsp; Number encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/character_encoding/">3.4 &nbsp; Character encoding *</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_data_structure/summary/">3.5 &nbsp; Summary</a></li>
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en/chapter_data_structure/number_encoding/index.html
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<title>3.3 Number Encoding * - Hello Algo</title>
<title>3.3 Number encoding * - Hello Algo</title>
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3.3 Number Encoding *
3.3 Number encoding *
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@@ -201,7 +201,13 @@
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繁體中文
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@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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@@ -524,7 +530,7 @@
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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@@ -642,7 +648,7 @@
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -805,7 +811,7 @@
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
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@@ -821,7 +827,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
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Chapter 3. Data Structures
Chapter 3. Data structures
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@@ -838,7 +844,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -889,7 +895,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
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<li class="md-nav__item">
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<span class="md-ellipsis">
3.3.1 &nbsp; Integer Encoding
3.3.1 &nbsp; Integer encoding
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<li class="md-nav__item">
<a href="#332-floating-point-number-encoding" class="md-nav__link">
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3.3.2 &nbsp; Floating-Point Number Encoding
3.3.2 &nbsp; Floating-point number encoding
</span>
</a>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
</span>
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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@@ -1290,7 +1296,7 @@
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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@@ -1402,7 +1408,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
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3.3.1 &nbsp; Integer Encoding
3.3.1 &nbsp; Integer encoding
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3.3.2 &nbsp; Floating-Point Number Encoding
3.3.2 &nbsp; Floating-point number encoding
</span>
</a>
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<!-- Page content -->
<h1 id="33-number-encoding">3.3 &nbsp; Number Encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
<h1 id="33-number-encoding">3.3 &nbsp; Number encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>In this book, chapters marked with an asterisk '*' are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>
</div>
<h2 id="331-integer-encoding">3.3.1 &nbsp; Integer Encoding<a class="headerlink" href="#331-integer-encoding" title="Permanent link">&para;</a></h2>
<h2 id="331-integer-encoding">3.3.1 &nbsp; Integer encoding<a class="headerlink" href="#331-integer-encoding" title="Permanent link">&para;</a></h2>
<p>In the table from the previous section, we observed that all integer types can represent one more negative number than positive numbers, such as the <code>byte</code> range of <span class="arithmatex">\([-128, 127]\)</span>. This phenomenon seems counterintuitive, and its underlying reason involves knowledge of sign-magnitude, one's complement, and two's complement encoding.</p>
<p>Firstly, it's important to note that <strong>numbers are stored in computers using the two's complement form</strong>. Before analyzing why this is the case, let's define these three encoding methods:</p>
<ul>
@@ -2084,8 +2090,8 @@
<li><strong>Two's complement</strong>: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding <span class="arithmatex">\(1\)</span> to their one's complement.</li>
</ul>
<p>The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:</p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between Sign-Magnitude, One's Complement, and Two's Complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> Figure 3-4 &nbsp; Conversions between Sign-Magnitude, One's Complement, and Two's Complement </p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between sign-magnitude, one's complement, and two's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> Figure 3-4 &nbsp; Conversions between sign-magnitude, one's complement, and two's complement </p>
<p>Although sign-magnitude is the most intuitive, it has limitations. For one, <strong>negative numbers in sign-magnitude cannot be directly used in calculations</strong>. For example, in sign-magnitude, calculating <span class="arithmatex">\(1 + (-2)\)</span> results in <span class="arithmatex">\(-3\)</span>, which is incorrect.</p>
<div class="arithmatex">\[
@@ -2139,7 +2145,7 @@
<p>It's important to note that this doesn't mean computers can only perform addition. <strong>By combining addition with basic logical operations, computers can execute a variety of other mathematical operations</strong>. For example, the subtraction <span class="arithmatex">\(a - b\)</span> can be translated into <span class="arithmatex">\(a + (-b)\)</span>; multiplication and division can be translated into multiple additions or subtractions.</p>
<p>We can now summarize the reason for using two's complement in computers: with two's complement representation, computers can use the same circuits and operations to handle both positive and negative number addition, eliminating the need for special hardware circuits for subtraction and avoiding the ambiguity of positive and negative zero. This greatly simplifies hardware design and enhances computational efficiency.</p>
<p>The design of two's complement is quite ingenious, and due to space constraints, we'll stop here. Interested readers are encouraged to explore further.</p>
<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-Point Number Encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-point number encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
<p>You might have noticed something intriguing: despite having the same length of 4 bytes, why does a <code>float</code> have a much larger range of values compared to an <code>int</code>? This seems counterintuitive, as one would expect the range to shrink for <code>float</code> since it needs to represent fractions.</p>
<p>In fact, <strong>this is due to the different representation method used by floating-point numbers (<code>float</code>)</strong>. Let's consider a 32-bit binary number as:</p>
<div class="arithmatex">\[
@@ -2166,8 +2172,8 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
(1 + \mathrm{N}) = &amp; (1 + \sum_{i=1}^{23} b_{23-i} \times 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
\]</div>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example Calculation of a float in IEEE 754 Standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> Figure 3-5 &nbsp; Example Calculation of a float in IEEE 754 Standard </p>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example calculation of a float in IEEE 754 standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> Figure 3-5 &nbsp; Example calculation of a float in IEEE 754 standard </p>
<p>Observing the diagram, given an example data <span class="arithmatex">\(\mathrm{S} = 0\)</span>, <span class="arithmatex">\(\mathrm{E} = 124\)</span>, <span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span>, we have:</p>
<div class="arithmatex">\[
@@ -2176,7 +2182,7 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
<p>Now we can answer the initial question: <strong>The representation of <code>float</code> includes an exponent bit, leading to a much larger range than <code>int</code></strong>. Based on the above calculation, the maximum positive number representable by <code>float</code> is approximately <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span>, and the minimum negative number is obtained by switching the sign bit.</p>
<p><strong>However, the trade-off for <code>float</code>'s expanded range is a sacrifice in precision</strong>. The integer type <code>int</code> uses all 32 bits to represent the number, with values evenly distributed; but due to the exponent bit, the larger the value of a <code>float</code>, the greater the difference between adjacent numbers.</p>
<p>As shown in the Table 3-2 , exponent bits <span class="arithmatex">\(E = 0\)</span> and <span class="arithmatex">\(E = 255\)</span> have special meanings, <strong>used to represent zero, infinity, <span class="arithmatex">\(\mathrm{NaN}\)</span>, etc.</strong></p>
<p align="center"> Table 3-2 &nbsp; Meaning of Exponent Bits </p>
<p align="center"> Table 3-2 &nbsp; Meaning of exponent bits </p>
<div class="center-table">
<table>
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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Chapter 4. Array and Linked List
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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Chapter 5. Stack and Queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
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<!-- Page content -->
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<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<h3 id="1-key-review">1. &nbsp; Key review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<ul>
<li>Data structures can be categorized from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory.</li>
<li>Common logical structures include linear, tree-like, and network structures. We generally classify data structures into linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. The implementation of hash tables may involve both linear and non-linear data structures.</li>
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@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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@@ -507,7 +513,7 @@
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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@@ -524,7 +530,7 @@
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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@@ -642,7 +648,7 @@
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Chapter 2. Complexity Analysis
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@@ -659,7 +665,7 @@
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
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Chapter 3. Data Structures
Chapter 3. Data structures
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@@ -819,7 +825,7 @@
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Chapter 3. Data Structures
Chapter 3. Data structures
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@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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@@ -996,7 +1002,7 @@
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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@@ -1034,7 +1040,7 @@
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4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1171,7 +1177,7 @@
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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Chapter 6. Hash Table
Chapter 6. Hash table
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@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_graph/graph_operations/index.html
+37 -31
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@@ -201,7 +201,13 @@
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中文
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繁體中文
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@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -996,7 +1002,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1040,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1171,7 +1177,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1363,7 +1369,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_graph/graph_traversal/index.html
+37 -31
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@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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en/chapter_graph/index.html
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
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<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -412,7 +418,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<title>6.3 Hash Algorithm - Hello Algo</title>
<title>6.3 Hash algorithm - Hello Algo</title>
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6.3 Hash Algorithm
6.3 Hash algorithm
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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6.3 Hash Algorithm
6.3 Hash algorithm
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<a href="#631-goals-of-hash-algorithms" class="md-nav__link">
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6.3.1 &nbsp; Goals of Hash Algorithms
6.3.1 &nbsp; Goals of hash algorithms
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6.3.2 &nbsp; Design of Hash Algorithms
6.3.2 &nbsp; Design of hash algorithms
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6.3.3 &nbsp; Common Hash Algorithms
6.3.3 &nbsp; Common hash algorithms
</span>
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<a href="#631-goals-of-hash-algorithms" class="md-nav__link">
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6.3.1 &nbsp; Goals of Hash Algorithms
6.3.1 &nbsp; Goals of hash algorithms
</span>
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6.3.2 &nbsp; Design of Hash Algorithms
6.3.2 &nbsp; Design of hash algorithms
</span>
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6.3.3 &nbsp; Common Hash Algorithms
6.3.3 &nbsp; Common hash algorithms
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<!-- Page content -->
<h1 id="63-hash-algorithms">6.3 &nbsp; Hash Algorithms<a class="headerlink" href="#63-hash-algorithms" title="Permanent link">&para;</a></h1>
<h1 id="63-hash-algorithms">6.3 &nbsp; Hash algorithms<a class="headerlink" href="#63-hash-algorithms" title="Permanent link">&para;</a></h1>
<p>The previous two sections introduced the working principle of hash tables and the methods to handle hash collisions. However, both open addressing and chaining can <strong>only ensure that the hash table functions normally when collisions occur, but cannot reduce the frequency of hash collisions</strong>.</p>
<p>If hash collisions occur too frequently, the performance of the hash table will deteriorate drastically. As shown in the Figure 6-8 , for a chaining hash table, in the ideal case, the key-value pairs are evenly distributed across the buckets, achieving optimal query efficiency; in the worst case, all key-value pairs are stored in the same bucket, degrading the time complexity to <span class="arithmatex">\(O(n)\)</span>.</p>
<p><a class="glightbox" href="../hash_algorithm.assets/hash_collision_best_worst_condition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Ideal and Worst Cases of Hash Collisions" class="animation-figure" src="../hash_algorithm.assets/hash_collision_best_worst_condition.png" /></a></p>
<p align="center"> Figure 6-8 &nbsp; Ideal and Worst Cases of Hash Collisions </p>
<p><a class="glightbox" href="../hash_algorithm.assets/hash_collision_best_worst_condition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Ideal and worst cases of hash collisions" class="animation-figure" src="../hash_algorithm.assets/hash_collision_best_worst_condition.png" /></a></p>
<p align="center"> Figure 6-8 &nbsp; Ideal and worst cases of hash collisions </p>
<p><strong>The distribution of key-value pairs is determined by the hash function</strong>. Recalling the steps of calculating a hash function, first compute the hash value, then modulo it by the array length:</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
</code></pre></div>
<p>Observing the above formula, when the hash table capacity <code>capacity</code> is fixed, <strong>the hash algorithm <code>hash()</code> determines the output value</strong>, thereby determining the distribution of key-value pairs in the hash table.</p>
<p>This means that, to reduce the probability of hash collisions, we should focus on the design of the hash algorithm <code>hash()</code>.</p>
<h2 id="631-goals-of-hash-algorithms">6.3.1 &nbsp; Goals of Hash Algorithms<a class="headerlink" href="#631-goals-of-hash-algorithms" title="Permanent link">&para;</a></h2>
<h2 id="631-goals-of-hash-algorithms">6.3.1 &nbsp; Goals of hash algorithms<a class="headerlink" href="#631-goals-of-hash-algorithms" title="Permanent link">&para;</a></h2>
<p>To achieve a "fast and stable" hash table data structure, hash algorithms should have the following characteristics:</p>
<ul>
<li><strong>Determinism</strong>: For the same input, the hash algorithm should always produce the same output. Only then can the hash table be reliable.</li>
<li><strong>High Efficiency</strong>: The process of computing the hash value should be fast enough. The smaller the computational overhead, the more practical the hash table.</li>
<li><strong>Uniform Distribution</strong>: The hash algorithm should ensure that key-value pairs are evenly distributed in the hash table. The more uniform the distribution, the lower the probability of hash collisions.</li>
<li><strong>High efficiency</strong>: The process of computing the hash value should be fast enough. The smaller the computational overhead, the more practical the hash table.</li>
<li><strong>Uniform distribution</strong>: The hash algorithm should ensure that key-value pairs are evenly distributed in the hash table. The more uniform the distribution, the lower the probability of hash collisions.</li>
</ul>
<p>In fact, hash algorithms are not only used to implement hash tables but are also widely applied in other fields.</p>
<ul>
<li><strong>Password Storage</strong>: To protect the security of user passwords, systems usually do not store the plaintext passwords but rather the hash values of the passwords. When a user enters a password, the system calculates the hash value of the input and compares it with the stored hash value. If they match, the password is considered correct.</li>
<li><strong>Data Integrity Check</strong>: The data sender can calculate the hash value of the data and send it along; the receiver can recalculate the hash value of the received data and compare it with the received hash value. If they match, the data is considered intact.</li>
<li><strong>Password storage</strong>: To protect the security of user passwords, systems usually do not store the plaintext passwords but rather the hash values of the passwords. When a user enters a password, the system calculates the hash value of the input and compares it with the stored hash value. If they match, the password is considered correct.</li>
<li><strong>Data integrity check</strong>: The data sender can calculate the hash value of the data and send it along; the receiver can recalculate the hash value of the received data and compare it with the received hash value. If they match, the data is considered intact.</li>
</ul>
<p>For cryptographic applications, to prevent reverse engineering such as deducing the original password from the hash value, hash algorithms need higher-level security features.</p>
<ul>
<li><strong>Unidirectionality</strong>: It should be impossible to deduce any information about the input data from the hash value.</li>
<li><strong>Collision Resistance</strong>: It should be extremely difficult to find two different inputs that produce the same hash value.</li>
<li><strong>Avalanche Effect</strong>: Minor changes in the input should lead to significant and unpredictable changes in the output.</li>
<li><strong>Collision resistance</strong>: It should be extremely difficult to find two different inputs that produce the same hash value.</li>
<li><strong>Avalanche effect</strong>: Minor changes in the input should lead to significant and unpredictable changes in the output.</li>
</ul>
<p>Note that <strong>"Uniform Distribution" and "Collision Resistance" are two separate concepts</strong>. Satisfying uniform distribution does not necessarily mean collision resistance. For example, under random input <code>key</code>, the hash function <code>key % 100</code> can produce a uniformly distributed output. However, this hash algorithm is too simple, and all <code>key</code> with the same last two digits will have the same output, making it easy to deduce a usable <code>key</code> from the hash value, thereby cracking the password.</p>
<h2 id="632-design-of-hash-algorithms">6.3.2 &nbsp; Design of Hash Algorithms<a class="headerlink" href="#632-design-of-hash-algorithms" title="Permanent link">&para;</a></h2>
<h2 id="632-design-of-hash-algorithms">6.3.2 &nbsp; Design of hash algorithms<a class="headerlink" href="#632-design-of-hash-algorithms" title="Permanent link">&para;</a></h2>
<p>The design of hash algorithms is a complex issue that requires consideration of many factors. However, for some less demanding scenarios, we can also design some simple hash algorithms.</p>
<ul>
<li><strong>Additive Hash</strong>: Add up the ASCII codes of each character in the input and use the total sum as the hash value.</li>
<li><strong>Multiplicative Hash</strong>: Utilize the non-correlation of multiplication, multiplying each round by a constant, accumulating the ASCII codes of each character into the hash value.</li>
<li><strong>XOR Hash</strong>: Accumulate the hash value by XORing each element of the input data.</li>
<li><strong>Rotating Hash</strong>: Accumulate the ASCII code of each character into a hash value, performing a rotation operation on the hash value before each accumulation.</li>
<li><strong>Additive hash</strong>: Add up the ASCII codes of each character in the input and use the total sum as the hash value.</li>
<li><strong>Multiplicative hash</strong>: Utilize the non-correlation of multiplication, multiplying each round by a constant, accumulating the ASCII codes of each character into the hash value.</li>
<li><strong>XOR hash</strong>: Accumulate the hash value by XORing each element of the input data.</li>
<li><strong>Rotating hash</strong>: Accumulate the ASCII code of each character into a hash value, performing a rotation operation on the hash value before each accumulation.</li>
</ul>
<div class="tabbed-set tabbed-alternate" data-tabs="1:14"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label><label for="__tabbed_1_14">Zig</label></div>
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@@ -2693,7 +2699,7 @@
\]</div>
<p>It is worth noting that if the <code>key</code> is guaranteed to be randomly and uniformly distributed, then choosing a prime number or a composite number as the modulus can both produce uniformly distributed hash values. However, when the distribution of <code>key</code> has some periodicity, modulo a composite number is more likely to result in clustering.</p>
<p>In summary, we usually choose a prime number as the modulus, and this prime number should be large enough to eliminate periodic patterns as much as possible, enhancing the robustness of the hash algorithm.</p>
<h2 id="633-common-hash-algorithms">6.3.3 &nbsp; Common Hash Algorithms<a class="headerlink" href="#633-common-hash-algorithms" title="Permanent link">&para;</a></h2>
<h2 id="633-common-hash-algorithms">6.3.3 &nbsp; Common hash algorithms<a class="headerlink" href="#633-common-hash-algorithms" title="Permanent link">&para;</a></h2>
<p>It is not hard to see that the simple hash algorithms mentioned above are quite "fragile" and far from reaching the design goals of hash algorithms. For example, since addition and XOR obey the commutative law, additive hash and XOR hash cannot distinguish strings with the same content but in different order, which may exacerbate hash collisions and cause security issues.</p>
<p>In practice, we usually use some standard hash algorithms, such as MD5, SHA-1, SHA-2, and SHA-3. They can map input data of any length to a fixed-length hash value.</p>
<p>Over the past century, hash algorithms have been in a continuous process of upgrading and optimization. Some researchers strive to improve the performance of hash algorithms, while others, including hackers, are dedicated to finding security issues in hash algorithms. The Table 6-2 shows hash algorithms commonly used in practical applications.</p>
@@ -2702,7 +2708,7 @@
<li>SHA-2 series, especially SHA-256, is one of the most secure hash algorithms to date, with no successful attacks reported, hence commonly used in various security applications and protocols.</li>
<li>SHA-3 has lower implementation costs and higher computational efficiency compared to SHA-2, but its current usage coverage is not as extensive as the SHA-2 series.</li>
</ul>
<p align="center"> Table 6-2 &nbsp; Common Hash Algorithms </p>
<p align="center"> Table 6-2 &nbsp; Common hash algorithms </p>
<div class="center-table">
<table>
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</tbody>
</table>
</div>
<h1 id="hash-values-in-data-structures">Hash Values in Data Structures<a class="headerlink" href="#hash-values-in-data-structures" title="Permanent link">&para;</a></h1>
<h1 id="hash-values-in-data-structures">Hash values in data structures<a class="headerlink" href="#hash-values-in-data-structures" title="Permanent link">&para;</a></h1>
<p>We know that the keys in a hash table can be of various data types such as integers, decimals, or strings. Programming languages usually provide built-in hash algorithms for these data types to calculate the bucket indices in the hash table. Taking Python as an example, we can use the <code>hash()</code> function to compute the hash values for various data types.</p>
<ul>
<li>The hash values of integers and booleans are their own values.</li>
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<title>Chapter 6.   Hash Table - Hello Algo</title>
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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<!-- Page content -->
<h1 id="chapter-6-hash-table">Chapter 6. &nbsp; Hash Table<a class="headerlink" href="#chapter-6-hash-table" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_hashing.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Hash Table" class="cover-image" src="../assets/covers/chapter_hashing.jpg" /></a></p>
<h1 id="chapter-6-hash-table">Chapter 6. &nbsp; Hash table<a class="headerlink" href="#chapter-6-hash-table" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_hashing.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Hash table" class="cover-image" src="../assets/covers/chapter_hashing.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>In the world of computing, a hash table is akin to an intelligent librarian.</p>
@@ -2012,9 +2018,9 @@
</div>
<h2 id="chapter-contents">Chapter Contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/hash_map/">6.1 &nbsp; Hash Table</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/hash_collision/">6.2 &nbsp; Hash Collision</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/hash_algorithm/">6.3 &nbsp; Hash Algorithm</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/hash_map/">6.1 &nbsp; Hash table</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/hash_collision/">6.2 &nbsp; Hash collision</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/hash_algorithm/">6.3 &nbsp; Hash algorithm</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_hashing/summary/">6.4 &nbsp; Summary</a></li>
</ul>
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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<li class="md-nav__item">
<a href="#1-key-review" class="md-nav__link">
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1. &nbsp; Key Review
1. &nbsp; Key review
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1. &nbsp; Key Review
1. &nbsp; Key review
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<!-- Page content -->
<h1 id="64-summary">6.4 &nbsp; Summary<a class="headerlink" href="#64-summary" title="Permanent link">&para;</a></h1>
<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<h3 id="1-key-review">1. &nbsp; Key review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<ul>
<li>Given an input <code>key</code>, a hash table can retrieve the corresponding <code>value</code> in <span class="arithmatex">\(O(1)\)</span> time, which is highly efficient.</li>
<li>Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table.</li>
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繁體中文
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@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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3.3 Number Encoding *
3.3 Number encoding *
</span>
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
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中文
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繁體中文
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@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<label class="md-nav__title" for="__nav_5">
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
</span>
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
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<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -996,7 +1002,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1040,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1171,7 +1177,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1363,7 +1369,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_heap/summary/index.html
+37 -31
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@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -996,7 +1002,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1040,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1171,7 +1177,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1363,7 +1369,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_heap/top_k/index.html
+37 -31
View File
@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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Chapter 5. Stack and Queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash table
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6.1 Hash table
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<title>1.1 Algorithms are Everywhere - Hello Algo</title>
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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中文
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.3 Hash Algorithm
6.3 Hash algorithm
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<!-- Page content -->
<h1 id="11-algorithms-are-everywhere">1.1 &nbsp; Algorithms are Everywhere<a class="headerlink" href="#11-algorithms-are-everywhere" title="Permanent link">&para;</a></h1>
<h1 id="11-algorithms-are-everywhere">1.1 &nbsp; Algorithms are everywhere<a class="headerlink" href="#11-algorithms-are-everywhere" title="Permanent link">&para;</a></h1>
<p>When we hear the word "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p>
<p>Before formally discussing algorithms, there's an interesting fact worth sharing: <strong>you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life</strong>. Here, I will give a few specific examples to prove this point.</p>
<p><strong>Example 1: Looking Up a Dictionary</strong>. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter <span class="arithmatex">\(r\)</span>. This is typically done in the following way:</p>
@@ -2033,8 +2039,8 @@
<li>Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.</li>
<li>Continue to repeat step <code>2.</code> until all cards are in order.</li>
</ol>
<p><a class="glightbox" href="../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Playing Cards Sorting Process" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></a></p>
<p align="center"> Figure 1-2 &nbsp; Playing Cards Sorting Process </p>
<p><a class="glightbox" href="../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Playing cards sorting process" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></a></p>
<p align="center"> Figure 1-2 &nbsp; Playing cards sorting process </p>
<p>The above method of organizing playing cards is essentially the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.</p>
<p><strong>Example 3: Making Change</strong>. Suppose we buy goods worth <span class="arithmatex">\(69\)</span> yuan at a supermarket and give the cashier <span class="arithmatex">\(100\)</span> yuan, then the cashier needs to give us <span class="arithmatex">\(31\)</span> yuan in change. They would naturally complete the thought process as shown below.</p>
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0.1 About This Book
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0.2 How to Read
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
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5.3 Double-ended Queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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<!-- Page content -->
<h1 id="chapter-1-introduction-to-algorithms">Chapter 1. &nbsp; Introduction to Algorithms<a class="headerlink" href="#chapter-1-introduction-to-algorithms" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_introduction.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="A first look at the algorithm" class="cover-image" src="../assets/covers/chapter_introduction.jpg" /></a></p>
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<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>A graceful maiden dances, intertwined with the data, her skirt swaying to the melody of algorithms.</p>
@@ -2012,8 +2018,8 @@
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<li><a href="https://www.hello-algo.com/en/chapter_introduction/algorithms_are_everywhere/">1.1 &nbsp; Algorithms are Everywhere</a></li>
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0.1 About This Book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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2.1 Algorithm Efficiency Assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
</span>
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3.4 Character Encoding *
3.4 Character encoding *
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4.2 Linked List
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4.4 Memory and Cache
4.4 Memory and cache
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</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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1.2 What is an Algorithm
1.2 What is an algorithm
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<a href="#121-definition-of-an-algorithm" class="md-nav__link">
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1.2.1 &nbsp; Definition of an Algorithm
1.2.1 &nbsp; Definition of an algorithm
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1.2.2 &nbsp; Definition of a Data Structure
1.2.2 &nbsp; Definition of a data structure
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<a href="#123-relationship-between-data-structures-and-algorithms" class="md-nav__link">
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1.2.3 &nbsp; Relationship Between Data Structures and Algorithms
1.2.3 &nbsp; Relationship between data structures and algorithms
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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1.2.1 &nbsp; Definition of an Algorithm
1.2.1 &nbsp; Definition of an algorithm
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1.2.2 &nbsp; Definition of a data structure
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1.2.3 &nbsp; Relationship Between Data Structures and Algorithms
1.2.3 &nbsp; Relationship between data structures and algorithms
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<!-- Page content -->
<h1 id="12-what-is-an-algorithm">1.2 &nbsp; What is an Algorithm<a class="headerlink" href="#12-what-is-an-algorithm" title="Permanent link">&para;</a></h1>
<h2 id="121-definition-of-an-algorithm">1.2.1 &nbsp; Definition of an Algorithm<a class="headerlink" href="#121-definition-of-an-algorithm" title="Permanent link">&para;</a></h2>
<h1 id="12-what-is-an-algorithm">1.2 &nbsp; What is an algorithm<a class="headerlink" href="#12-what-is-an-algorithm" title="Permanent link">&para;</a></h1>
<h2 id="121-definition-of-an-algorithm">1.2.1 &nbsp; Definition of an algorithm<a class="headerlink" href="#121-definition-of-an-algorithm" title="Permanent link">&para;</a></h2>
<p>An "algorithm" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:</p>
<ul>
<li>The problem is clearly defined, including unambiguous definitions of input and output.</li>
<li>The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space.</li>
<li>Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions.</li>
</ul>
<h2 id="122-definition-of-a-data-structure">1.2.2 &nbsp; Definition of a Data Structure<a class="headerlink" href="#122-definition-of-a-data-structure" title="Permanent link">&para;</a></h2>
<h2 id="122-definition-of-a-data-structure">1.2.2 &nbsp; Definition of a data structure<a class="headerlink" href="#122-definition-of-a-data-structure" title="Permanent link">&para;</a></h2>
<p>A "data structure" is a way of organizing and storing data in a computer, with the following design goals:</p>
<ul>
<li>Minimize space occupancy to save computer memory.</li>
@@ -2108,7 +2114,7 @@
<li>Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.</li>
<li>Graphs, compared to linked lists, provide richer logical information but require more memory space.</li>
</ul>
<h2 id="123-relationship-between-data-structures-and-algorithms">1.2.3 &nbsp; Relationship Between Data Structures and Algorithms<a class="headerlink" href="#123-relationship-between-data-structures-and-algorithms" title="Permanent link">&para;</a></h2>
<h2 id="123-relationship-between-data-structures-and-algorithms">1.2.3 &nbsp; Relationship between data structures and algorithms<a class="headerlink" href="#123-relationship-between-data-structures-and-algorithms" title="Permanent link">&para;</a></h2>
<p>As shown in the Figure 1-4 , data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:</p>
<ul>
<li>Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.</li>
@@ -2123,7 +2129,7 @@
<p align="center"> Figure 1-5 &nbsp; Assembling blocks </p>
<p>The detailed correspondence between the two is shown in the Table 1-1 .</p>
<p align="center"> Table 1-1 &nbsp; Comparing Data Structures and Algorithms to Building Blocks </p>
<p align="center"> Table 1-1 &nbsp; Comparing data structures and algorithms to building blocks </p>
<div class="center-table">
<table>
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<title>0.1 About This Book - Hello Algo</title>
<title>0.1 About this book - Hello Algo</title>
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0.1 About This Book
0.1 About this book
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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繁體中文
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0.1 About This Book
0.1 About this book
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0.1 About This Book
0.1 About this book
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<a href="#011-target-audience" class="md-nav__link">
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0.1.1 &nbsp; Target Audience
0.1.1 &nbsp; Target audience
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0.1.2 &nbsp; Content Structure
0.1.2 &nbsp; Content structure
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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Chapter 1. Introduction to Algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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<!-- Page content -->
<h1 id="01-about-this-book">0.1 &nbsp; About This Book<a class="headerlink" href="#01-about-this-book" title="Permanent link">&para;</a></h1>
<h1 id="01-about-this-book">0.1 &nbsp; About this book<a class="headerlink" href="#01-about-this-book" title="Permanent link">&para;</a></h1>
<p>This open-source project aims to create a free, and beginner-friendly crash course on data structures and algorithms.</p>
<ul>
<li>Using animated illustrations, it delivers structured insights into data structures and algorithmic concepts, ensuring comprehensibility and a smooth learning curve.</li>
<li>Run code with just one click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages.</li>
<li>Readers are encouraged to engage with each other in the discussion area for each section, questions and comments are usually answered within two days.</li>
</ul>
<h2 id="011-target-audience">0.1.1 &nbsp; Target Audience<a class="headerlink" href="#011-target-audience" title="Permanent link">&para;</a></h2>
<h2 id="011-target-audience">0.1.1 &nbsp; Target audience<a class="headerlink" href="#011-target-audience" title="Permanent link">&para;</a></h2>
<p>If you are new to algorithms with limited exposure, or you have accumulated some experience in algorithms, but you only have a vague understanding of data structures and algorithms, and you are constantly jumping between "yep" and "hmm", then this book is for you!</p>
<p>If you have already accumulated a certain amount of problem-solving experience, and are familiar with most types of problems, then this book can help you review and organize your algorithm knowledge system. The repository's source code can be used as a "problem-solving toolkit" or an "algorithm cheat sheet".</p>
<p>If you are an algorithm expert, we look forward to receiving your valuable suggestions, or <a href="https://www.hello-algo.com/chapter_appendix/contribution/">join us and collaborate</a>.</p>
@@ -2103,15 +2109,15 @@
<p class="admonition-title">Prerequisites</p>
<p>You should know how to write and read simple code in at least one programming language.</p>
</div>
<h2 id="012-content-structure">0.1.2 &nbsp; Content Structure<a class="headerlink" href="#012-content-structure" title="Permanent link">&para;</a></h2>
<h2 id="012-content-structure">0.1.2 &nbsp; Content structure<a class="headerlink" href="#012-content-structure" title="Permanent link">&para;</a></h2>
<p>The main content of the book is shown in the following figure.</p>
<ul>
<li><strong>Complexity Analysis</strong>: explores aspects and methods for evaluating data structures and algorithms. Covers methods of deriving time complexity and space complexity, along with common types and examples.</li>
<li><strong>Data Structures</strong>: focuses on fundamental data types, classification methods, definitions, pros and cons, common operations, types, applications, and implementation methods of data structures such as array, linked list, stack, queue, hash table, tree, heap, graph, etc.</li>
<li><strong>Complexity analysis</strong>: explores aspects and methods for evaluating data structures and algorithms. Covers methods of deriving time complexity and space complexity, along with common types and examples.</li>
<li><strong>Data structures</strong>: focuses on fundamental data types, classification methods, definitions, pros and cons, common operations, types, applications, and implementation methods of data structures such as array, linked list, stack, queue, hash table, tree, heap, graph, etc.</li>
<li><strong>Algorithms</strong>: defines algorithms, discusses their pros and cons, efficiency, application scenarios, problem-solving steps, and includes sample questions for various algorithms such as search, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.</li>
</ul>
<p><a class="glightbox" href="../about_the_book.assets/hello_algo_mindmap.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Main Content of the Book" class="animation-figure" src="../about_the_book.assets/hello_algo_mindmap.png" /></a></p>
<p align="center"> Figure 0-1 &nbsp; Main Content of the Book </p>
<p><a class="glightbox" href="../about_the_book.assets/hello_algo_mindmap.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Main content of the book" class="animation-figure" src="../about_the_book.assets/hello_algo_mindmap.png" /></a></p>
<p align="center"> Figure 0-1 &nbsp; Main content of the book </p>
<h2 id="013-acknowledgements">0.1.3 &nbsp; Acknowledgements<a class="headerlink" href="#013-acknowledgements" title="Permanent link">&para;</a></h2>
<p>This book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub: krahets, codingonion, nuomi1, Gonglja, Reanon, justin-tse, danielsss, hpstory, S-N-O-R-L-A-X, night-cruise, msk397, gvenusleo, RiverTwilight, gyt95, zhuoqinyue, Zuoxun, Xia-Sang, mingXta, FangYuan33, GN-Yu, IsChristina, xBLACKICEx, guowei-gong, Cathay-Chen, mgisr, JoseHung, qualifier1024, pengchzn, Guanngxu, longsizhuo, L-Super, what-is-me, yuan0221, lhxsm, Slone123c, WSL0809, longranger2, theNefelibatas, xiongsp, JeffersonHuang, hongyun-robot, K3v123, yuelinxin, a16su, gaofer, malone6, Wonderdch, xjr7670, DullSword, Horbin-Magician, NI-SW, reeswell, XC-Zero, XiaChuerwu, yd-j, iron-irax, huawuque404, MolDuM, Nigh, KorsChen, foursevenlove, 52coder, bubble9um, youshaoXG, curly210102, gltianwen, fanchenggang, Transmigration-zhou, FloranceYeh, FreddieLi, ShiMaRing, lipusheng, Javesun99, JackYang-hellobobo, shanghai-Jerry, 0130w, Keynman, psychelzh, logan-qiu, ZnYang2018, MwumLi, 1ch0, Phoenix0415, qingpeng9802, Richard-Zhang1019, QiLOL, Suremotoo, Turing-1024-Lee, Evilrabbit520, GaochaoZhu, ZJKung, linzeyan, hezhizhen, ZongYangL, beintentional, czruby, coderlef, dshlstarr, szu17dmy, fbigm, gledfish, hts0000, boloboloda, iStig, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, liuxjerry, lucaswangdev, lyl625760, chadyi, noobcodemaker, selear, siqyka, syd168, 4yDX3906, tao363, wangwang105, weibk, yabo083, yi427, yishangzhang, zhouLion, baagod, ElaBosak233, xb534, luluxia, yanedie, thomasq0, YangXuanyi and th1nk3r-ing.</p>
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中文
简体中文
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繁體中文
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 3. Data Structures
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</div>
<h2 id="chapter-contents">Chapter Contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_preface/about_the_book/">0.1 &nbsp; About This Book</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_preface/suggestions/">0.2 &nbsp; How to Read</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_preface/about_the_book/">0.1 &nbsp; About this book</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_preface/suggestions/">0.2 &nbsp; How to read</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_preface/summary/">0.3 &nbsp; Summary</a></li>
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0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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0.2 How to Read
0.2 How to read
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0.2.2 &nbsp; Efficient learning via animated illustrations
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0.2.3 &nbsp; Deepen understanding through coding practice
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1471,7 +1477,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2046,7 +2052,7 @@
<li class="md-nav__item">
<a href="#021-writing-conventions" class="md-nav__link">
<span class="md-ellipsis">
0.2.1 &nbsp; Writing Conventions
0.2.1 &nbsp; Writing conventions
</span>
</a>
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<li class="md-nav__item">
<a href="#022-efficient-learning-via-animated-illustrations" class="md-nav__link">
<span class="md-ellipsis">
0.2.2 &nbsp; Efficient Learning via Animated Illustrations
0.2.2 &nbsp; Efficient learning via animated illustrations
</span>
</a>
@@ -2064,7 +2070,7 @@
<li class="md-nav__item">
<a href="#023-deepen-understanding-through-coding-practice" class="md-nav__link">
<span class="md-ellipsis">
0.2.3 &nbsp; Deepen Understanding through Coding Practice
0.2.3 &nbsp; Deepen understanding through coding practice
</span>
</a>
@@ -2073,7 +2079,7 @@
<li class="md-nav__item">
<a href="#024-learning-together-in-discussion" class="md-nav__link">
<span class="md-ellipsis">
0.2.4 &nbsp; Learning Together in Discussion
0.2.4 &nbsp; Learning together in discussion
</span>
</a>
@@ -2082,7 +2088,7 @@
<li class="md-nav__item">
<a href="#025-algorithm-learning-path" class="md-nav__link">
<span class="md-ellipsis">
0.2.5 &nbsp; Algorithm Learning Path
0.2.5 &nbsp; Algorithm learning path
</span>
</a>
@@ -2124,12 +2130,12 @@
<!-- Page content -->
<h1 id="02-how-to-read">0.2 &nbsp; How to Read<a class="headerlink" href="#02-how-to-read" title="Permanent link">&para;</a></h1>
<h1 id="02-how-to-read">0.2 &nbsp; How to read<a class="headerlink" href="#02-how-to-read" title="Permanent link">&para;</a></h1>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>For the best reading experience, it is recommended that you read through this section.</p>
</div>
<h2 id="021-writing-conventions">0.2.1 &nbsp; Writing Conventions<a class="headerlink" href="#021-writing-conventions" title="Permanent link">&para;</a></h2>
<h2 id="021-writing-conventions">0.2.1 &nbsp; Writing conventions<a class="headerlink" href="#021-writing-conventions" title="Permanent link">&para;</a></h2>
<ul>
<li>Chapters marked with '*' after the title are optional and contain relatively challenging content. If you are short on time, it is advisable to skip them.</li>
<li>Technical terms will be in boldface (in the print and PDF versions) or underlined (in the web version), for instance, <u>array</u>. It's advisable to familiarize yourself with these for better comprehension of technical texts.</li>
@@ -2141,7 +2147,7 @@
<div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="sd">&quot;&quot;&quot;Header comments for labeling functions, classes, test samples, etc&quot;&quot;&quot;</span><span class="s2">&quot;</span>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="sd">&quot;&quot;&quot;Header comments for labeling functions, classes, test samples, etc&quot;&quot;&quot;</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Comments for explaining details</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
@@ -2283,18 +2289,18 @@
</div>
</div>
</div>
<h2 id="022-efficient-learning-via-animated-illustrations">0.2.2 &nbsp; Efficient Learning via Animated Illustrations<a class="headerlink" href="#022-efficient-learning-via-animated-illustrations" title="Permanent link">&para;</a></h2>
<h2 id="022-efficient-learning-via-animated-illustrations">0.2.2 &nbsp; Efficient learning via animated illustrations<a class="headerlink" href="#022-efficient-learning-via-animated-illustrations" title="Permanent link">&para;</a></h2>
<p>Compared with text, videos and pictures have a higher density of information and are more structured, making them easier to understand. In this book, <strong>key and difficult concepts are mainly presented through animations and illustrations</strong>, with text serving as explanations and supplements.</p>
<p>When encountering content with animations or illustrations as shown in the Figure 0-2 , <strong>prioritize understanding the figure, with text as supplementary</strong>, integrating both for a comprehensive understanding.</p>
<p><a class="glightbox" href="../../index.assets/animation.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Animated Illustration Example" class="animation-figure" src="../../index.assets/animation.gif" /></a></p>
<p align="center"> Figure 0-2 &nbsp; Animated Illustration Example </p>
<p><a class="glightbox" href="../../index.assets/animation.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Animated illustration example" class="animation-figure" src="../../index.assets/animation.gif" /></a></p>
<p align="center"> Figure 0-2 &nbsp; Animated illustration example </p>
<h2 id="023-deepen-understanding-through-coding-practice">0.2.3 &nbsp; Deepen Understanding through Coding Practice<a class="headerlink" href="#023-deepen-understanding-through-coding-practice" title="Permanent link">&para;</a></h2>
<h2 id="023-deepen-understanding-through-coding-practice">0.2.3 &nbsp; Deepen understanding through coding practice<a class="headerlink" href="#023-deepen-understanding-through-coding-practice" title="Permanent link">&para;</a></h2>
<p>The source code of this book is hosted on the <a href="https://github.com/krahets/hello-algo">GitHub Repository</a>. As shown in the Figure 0-3 , <strong>the source code comes with test examples and can be executed with just a single click</strong>.</p>
<p>If time permits, <strong>it's recommended to type out the code yourself</strong>. If pressed for time, at least read and run all the codes.</p>
<p>Compared to just reading code, writing code often yields more learning. <strong>Learning by doing is the real way to learn.</strong></p>
<p><a class="glightbox" href="../../index.assets/running_code.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Running Code Example" class="animation-figure" src="../../index.assets/running_code.gif" /></a></p>
<p align="center"> Figure 0-3 &nbsp; Running Code Example </p>
<p><a class="glightbox" href="../../index.assets/running_code.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Running code example" class="animation-figure" src="../../index.assets/running_code.gif" /></a></p>
<p align="center"> Figure 0-3 &nbsp; Running code example </p>
<p>Setting up to run the code involves three main steps.</p>
<p><strong>Step 1: Install a local programming environment</strong>. Follow the <a href="https://www.hello-algo.com/chapter_appendix/installation/">tutorial</a> in the appendix for installation, or skip this step if already installed.</p>
@@ -2303,20 +2309,20 @@
<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>git<span class="w"> </span>clone<span class="w"> </span>https://github.com/krahets/hello-algo.git
</code></pre></div>
<p>Alternatively, you can also click the "Download ZIP" button at the location shown in the Figure 0-4 to directly download the code as a compressed ZIP file. Then, you can simply extract it locally.</p>
<p><a class="glightbox" href="../suggestions.assets/download_code.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Cloning Repository and Downloading Code" class="animation-figure" src="../suggestions.assets/download_code.png" /></a></p>
<p align="center"> Figure 0-4 &nbsp; Cloning Repository and Downloading Code </p>
<p><a class="glightbox" href="../suggestions.assets/download_code.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Cloning repository and downloading code" class="animation-figure" src="../suggestions.assets/download_code.png" /></a></p>
<p align="center"> Figure 0-4 &nbsp; Cloning repository and downloading code </p>
<p><strong>Step 3: Run the source code</strong>. As shown in the Figure 0-5 , for the code block labeled with the file name at the top, we can find the corresponding source code file in the <code>codes</code> folder of the repository. These files can be executed with a single click, which will help you save unnecessary debugging time and allow you to focus on learning.</p>
<p><a class="glightbox" href="../suggestions.assets/code_md_to_repo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Code Block and Corresponding Source Code File" class="animation-figure" src="../suggestions.assets/code_md_to_repo.png" /></a></p>
<p align="center"> Figure 0-5 &nbsp; Code Block and Corresponding Source Code File </p>
<p><a class="glightbox" href="../suggestions.assets/code_md_to_repo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Code block and corresponding source code file" class="animation-figure" src="../suggestions.assets/code_md_to_repo.png" /></a></p>
<p align="center"> Figure 0-5 &nbsp; Code block and corresponding source code file </p>
<h2 id="024-learning-together-in-discussion">0.2.4 &nbsp; Learning Together in Discussion<a class="headerlink" href="#024-learning-together-in-discussion" title="Permanent link">&para;</a></h2>
<h2 id="024-learning-together-in-discussion">0.2.4 &nbsp; Learning together in discussion<a class="headerlink" href="#024-learning-together-in-discussion" title="Permanent link">&para;</a></h2>
<p>While reading this book, please don't skip over the points that you didn't learn. <strong>Feel free to post your questions in the comment section</strong>. We will be happy to answer them and can usually respond within two days.</p>
<p>As illustrated in the Figure 0-6 , each chapter features a comment section at the bottom. I encourage you to pay attention to these comments. They not only expose you to others' encountered problems, aiding in identifying knowledge gaps and sparking deeper contemplation, but also invite you to generously contribute by answering fellow readers' inquiries, sharing insights, and fostering mutual improvement.</p>
<p><a class="glightbox" href="../../index.assets/comment.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Comment Section Example" class="animation-figure" src="../../index.assets/comment.gif" /></a></p>
<p align="center"> Figure 0-6 &nbsp; Comment Section Example </p>
<p><a class="glightbox" href="../../index.assets/comment.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Comment section example" class="animation-figure" src="../../index.assets/comment.gif" /></a></p>
<p align="center"> Figure 0-6 &nbsp; Comment section example </p>
<h2 id="025-algorithm-learning-path">0.2.5 &nbsp; Algorithm Learning Path<a class="headerlink" href="#025-algorithm-learning-path" title="Permanent link">&para;</a></h2>
<h2 id="025-algorithm-learning-path">0.2.5 &nbsp; Algorithm learning path<a class="headerlink" href="#025-algorithm-learning-path" title="Permanent link">&para;</a></h2>
<p>Overall, the journey of mastering data structures and algorithms can be divided into three stages:</p>
<ol>
<li><strong>Stage 1: Introduction to algorithms</strong>. We need to familiarize ourselves with the characteristics and usage of various data structures and learn about the principles, processes, uses, and efficiency of different algorithms.</li>
@@ -2324,8 +2330,8 @@
<li><strong>Stage 3: Building the knowledge system</strong>. In terms of learning, we can read algorithm column articles, solution frameworks, and algorithm textbooks to continuously enrich the knowledge system. In terms of practicing, we can try advanced strategies, such as categorizing by topic, multiple solutions for a single problem, and one solution for multiple problems, etc. Insights on these strategies can be found in various communities.</li>
</ol>
<p>As shown in the Figure 0-7 , this book mainly covers “Stage 1,” aiming to help you more efficiently embark on Stages 2 and 3.</p>
<p><a class="glightbox" href="../suggestions.assets/learning_route.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Algorithm Learning Path" class="animation-figure" src="../suggestions.assets/learning_route.png" /></a></p>
<p align="center"> Figure 0-7 &nbsp; Algorithm Learning Path </p>
<p><a class="glightbox" href="../suggestions.assets/learning_route.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Algorithm learning path" class="animation-figure" src="../suggestions.assets/learning_route.png" /></a></p>
<p align="center"> Figure 0-7 &nbsp; Algorithm learning path </p>
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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2.1 Algorithm Efficiency Assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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0.1 About This Book
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0.2 How to Read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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5.3 Double-ended Queue
5.3 Double-ended queue
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5.3.1 &nbsp; Common operations in double-ended queue
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5.3.2 &nbsp; Implementing a Double-Ended Queue *
5.3.2 &nbsp; Implementing a double-ended queue *
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1. &nbsp; Implementation Based on Doubly Linked List
1. &nbsp; Implementation based on doubly linked list
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2. &nbsp; Implementation based on array
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5.3.3 &nbsp; Applications of Double-Ended Queue
5.3.3 &nbsp; Applications of double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2052,7 +2058,7 @@
<li class="md-nav__item">
<a href="#531-common-operations-in-double-ended-queue" class="md-nav__link">
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5.3.1 &nbsp; Common Operations in Double-Ended Queue
5.3.1 &nbsp; Common operations in double-ended queue
</span>
</a>
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<li class="md-nav__item">
<a href="#532-implementing-a-double-ended-queue" class="md-nav__link">
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5.3.2 &nbsp; Implementing a Double-Ended Queue *
5.3.2 &nbsp; Implementing a double-ended queue *
</span>
</a>
<nav class="md-nav" aria-label="5.3.2   Implementing a Double-Ended Queue *">
<nav class="md-nav" aria-label="5.3.2   Implementing a double-ended queue *">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#1-implementation-based-on-doubly-linked-list" class="md-nav__link">
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1. &nbsp; Implementation Based on Doubly Linked List
1. &nbsp; Implementation based on doubly linked list
</span>
</a>
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<li class="md-nav__item">
<a href="#2-implementation-based-on-array" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Implementation Based on Array
2. &nbsp; Implementation based on array
</span>
</a>
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<li class="md-nav__item">
<a href="#533-applications-of-double-ended-queue" class="md-nav__link">
<span class="md-ellipsis">
5.3.3 &nbsp; Applications of Double-Ended Queue
5.3.3 &nbsp; Applications of double-ended queue
</span>
</a>
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<!-- Page content -->
<h1 id="53-double-ended-queue">5.3 &nbsp; Double-Ended Queue<a class="headerlink" href="#53-double-ended-queue" title="Permanent link">&para;</a></h1>
<h1 id="53-double-ended-queue">5.3 &nbsp; Double-ended queue<a class="headerlink" href="#53-double-ended-queue" title="Permanent link">&para;</a></h1>
<p>In a queue, we can only delete elements from the head or add elements to the tail. As shown in the following diagram, a "double-ended queue (deque)" offers more flexibility, allowing the addition or removal of elements at both the head and the tail.</p>
<p><a class="glightbox" href="../deque.assets/deque_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Operations in Double-Ended Queue" class="animation-figure" src="../deque.assets/deque_operations.png" /></a></p>
<p align="center"> Figure 5-7 &nbsp; Operations in Double-Ended Queue </p>
<p><a class="glightbox" href="../deque.assets/deque_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Operations in double-ended queue" class="animation-figure" src="../deque.assets/deque_operations.png" /></a></p>
<p align="center"> Figure 5-7 &nbsp; Operations in double-ended queue </p>
<h2 id="531-common-operations-in-double-ended-queue">5.3.1 &nbsp; Common Operations in Double-Ended Queue<a class="headerlink" href="#531-common-operations-in-double-ended-queue" title="Permanent link">&para;</a></h2>
<h2 id="531-common-operations-in-double-ended-queue">5.3.1 &nbsp; Common operations in double-ended queue<a class="headerlink" href="#531-common-operations-in-double-ended-queue" title="Permanent link">&para;</a></h2>
<p>The common operations in a double-ended queue are listed below, and the names of specific methods depend on the programming language used.</p>
<p align="center"> Table 5-3 &nbsp; Efficiency of Double-Ended Queue Operations </p>
<p align="center"> Table 5-3 &nbsp; Efficiency of double-ended queue operations </p>
<div class="center-table">
<table>
@@ -2486,9 +2492,9 @@
<summary>Visualizing Code</summary>
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</details>
<h2 id="532-implementing-a-double-ended-queue">5.3.2 &nbsp; Implementing a Double-Ended Queue *<a class="headerlink" href="#532-implementing-a-double-ended-queue" title="Permanent link">&para;</a></h2>
<h2 id="532-implementing-a-double-ended-queue">5.3.2 &nbsp; Implementing a double-ended queue *<a class="headerlink" href="#532-implementing-a-double-ended-queue" title="Permanent link">&para;</a></h2>
<p>The implementation of a double-ended queue is similar to that of a regular queue, it can be based on either a linked list or an array as the underlying data structure.</p>
<h3 id="1-implementation-based-on-doubly-linked-list">1. &nbsp; Implementation Based on Doubly Linked List<a class="headerlink" href="#1-implementation-based-on-doubly-linked-list" title="Permanent link">&para;</a></h3>
<h3 id="1-implementation-based-on-doubly-linked-list">1. &nbsp; Implementation based on doubly linked list<a class="headerlink" href="#1-implementation-based-on-doubly-linked-list" title="Permanent link">&para;</a></h3>
<p>Recall from the previous section that we used a regular singly linked list to implement a queue, as it conveniently allows for deleting from the head (corresponding to the dequeue operation) and adding new elements after the tail (corresponding to the enqueue operation).</p>
<p>For a double-ended queue, both the head and the tail can perform enqueue and dequeue operations. In other words, a double-ended queue needs to implement operations in the opposite direction as well. For this, we use a "doubly linked list" as the underlying data structure of the double-ended queue.</p>
<p>As shown in the Figure 5-8 , we treat the head and tail nodes of the doubly linked list as the front and rear of the double-ended queue, respectively, and implement the functionality to add and remove nodes at both ends.</p>
@@ -4245,7 +4251,7 @@
</div>
</div>
</div>
<h3 id="2-implementation-based-on-array">2. &nbsp; Implementation Based on Array<a class="headerlink" href="#2-implementation-based-on-array" title="Permanent link">&para;</a></h3>
<h3 id="2-implementation-based-on-array">2. &nbsp; Implementation based on array<a class="headerlink" href="#2-implementation-based-on-array" title="Permanent link">&para;</a></h3>
<p>As shown in the Figure 5-9 , similar to implementing a queue with an array, we can also use a circular array to implement a double-ended queue.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:5"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">ArrayDeque</label><label for="__tabbed_4_2">pushLast()</label><label for="__tabbed_4_3">pushFirst()</label><label for="__tabbed_4_4">popLast()</label><label for="__tabbed_4_5">popFirst()</label></div>
<div class="tabbed-content">
@@ -5556,7 +5562,7 @@
</div>
</div>
</div>
<h2 id="533-applications-of-double-ended-queue">5.3.3 &nbsp; Applications of Double-Ended Queue<a class="headerlink" href="#533-applications-of-double-ended-queue" title="Permanent link">&para;</a></h2>
<h2 id="533-applications-of-double-ended-queue">5.3.3 &nbsp; Applications of double-ended queue<a class="headerlink" href="#533-applications-of-double-ended-queue" title="Permanent link">&para;</a></h2>
<p>The double-ended queue combines the logic of both stacks and queues, <strong>thus, it can implement all their respective use cases while offering greater flexibility</strong>.</p>
<p>We know that software's "undo" feature is typically implemented using a stack: the system <code>pushes</code> each change operation onto the stack and then <code>pops</code> to implement undoing. However, considering the limitations of system resources, software often restricts the number of undo steps (for example, only allowing the last 50 steps). When the stack length exceeds 50, the software needs to perform a deletion operation at the bottom of the stack (the front of the queue). <strong>But a regular stack cannot perform this function, where a double-ended queue becomes necessary</strong>. Note that the core logic of "undo" still follows the Last-In-First-Out principle of a stack, but a double-ended queue can more flexibly implement some additional logic.</p>
en/chapter_stack_and_queue/index.html
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<title>Chapter 5.   Stack and Queue - Hello Algo</title>
<title>Chapter 5.   Stack and queue - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
Chapter 5. &nbsp; Stack and Queue
Chapter 5. &nbsp; Stack and queue
</span>
</div>
@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="true">
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1327,7 +1333,7 @@
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2003,8 +2009,8 @@
<!-- Page content -->
<h1 id="chapter-5-stack-and-queue">Chapter 5. &nbsp; Stack and Queue<a class="headerlink" href="#chapter-5-stack-and-queue" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_stack_and_queue.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Stack and Queue" class="cover-image" src="../assets/covers/chapter_stack_and_queue.jpg" /></a></p>
<h1 id="chapter-5-stack-and-queue">Chapter 5. &nbsp; Stack and queue<a class="headerlink" href="#chapter-5-stack-and-queue" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_stack_and_queue.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Stack and queue" class="cover-image" src="../assets/covers/chapter_stack_and_queue.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>A stack is like cats placed on top of each other, while a queue is like cats lined up one by one.</p>
@@ -2014,7 +2020,7 @@
<ul>
<li><a href="https://www.hello-algo.com/en/chapter_stack_and_queue/stack/">5.1 &nbsp; Stack</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_stack_and_queue/queue/">5.2 &nbsp; Queue</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_stack_and_queue/deque/">5.3 &nbsp; Double-ended Queue</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_stack_and_queue/deque/">5.3 &nbsp; Double-ended queue</a></li>
<li><a href="https://www.hello-algo.com/en/chapter_stack_and_queue/summary/">5.4 &nbsp; Summary</a></li>
</ul>
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -996,7 +1002,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1040,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1157,7 +1163,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1173,7 +1179,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1234,7 +1240,7 @@
<li class="md-nav__item">
<a href="#511-common-operations-on-stack" class="md-nav__link">
<span class="md-ellipsis">
5.1.1 &nbsp; Common Operations on Stack
5.1.1 &nbsp; Common operations on stack
</span>
</a>
@@ -1243,26 +1249,26 @@
<li class="md-nav__item">
<a href="#512-implementing-a-stack" class="md-nav__link">
<span class="md-ellipsis">
5.1.2 &nbsp; Implementing a Stack
5.1.2 &nbsp; Implementing a stack
</span>
</a>
<nav class="md-nav" aria-label="5.1.2   Implementing a Stack">
<nav class="md-nav" aria-label="5.1.2   Implementing a stack">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#1-implementation-based-on-linked-list" class="md-nav__link">
<a href="#1-implementation-based-on-a-linked-list" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Implementation Based on Linked List
1. &nbsp; Implementation based on a linked list
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#2-implementation-based-on-array" class="md-nav__link">
<a href="#2-implementation-based-on-an-array" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Implementation Based on Array
2. &nbsp; Implementation based on an array
</span>
</a>
@@ -1276,7 +1282,7 @@
<li class="md-nav__item">
<a href="#513-comparison-of-the-two-implementations" class="md-nav__link">
<span class="md-ellipsis">
5.1.3 &nbsp; Comparison of the Two Implementations
5.1.3 &nbsp; Comparison of the two implementations
</span>
</a>
@@ -1285,7 +1291,7 @@
<li class="md-nav__item">
<a href="#514-typical-applications-of-stack" class="md-nav__link">
<span class="md-ellipsis">
5.1.4 &nbsp; Typical Applications of Stack
5.1.4 &nbsp; Typical applications of stack
</span>
</a>
@@ -1332,7 +1338,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1411,7 +1417,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1427,7 +1433,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1444,7 +1450,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1465,7 +1471,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1486,7 +1492,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2061,7 +2067,7 @@
<li class="md-nav__item">
<a href="#511-common-operations-on-stack" class="md-nav__link">
<span class="md-ellipsis">
5.1.1 &nbsp; Common Operations on Stack
5.1.1 &nbsp; Common operations on stack
</span>
</a>
@@ -2070,26 +2076,26 @@
<li class="md-nav__item">
<a href="#512-implementing-a-stack" class="md-nav__link">
<span class="md-ellipsis">
5.1.2 &nbsp; Implementing a Stack
5.1.2 &nbsp; Implementing a stack
</span>
</a>
<nav class="md-nav" aria-label="5.1.2   Implementing a Stack">
<nav class="md-nav" aria-label="5.1.2   Implementing a stack">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#1-implementation-based-on-linked-list" class="md-nav__link">
<a href="#1-implementation-based-on-a-linked-list" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Implementation Based on Linked List
1. &nbsp; Implementation based on a linked list
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#2-implementation-based-on-array" class="md-nav__link">
<a href="#2-implementation-based-on-an-array" class="md-nav__link">
<span class="md-ellipsis">
2. &nbsp; Implementation Based on Array
2. &nbsp; Implementation based on an array
</span>
</a>
@@ -2103,7 +2109,7 @@
<li class="md-nav__item">
<a href="#513-comparison-of-the-two-implementations" class="md-nav__link">
<span class="md-ellipsis">
5.1.3 &nbsp; Comparison of the Two Implementations
5.1.3 &nbsp; Comparison of the two implementations
</span>
</a>
@@ -2112,7 +2118,7 @@
<li class="md-nav__item">
<a href="#514-typical-applications-of-stack" class="md-nav__link">
<span class="md-ellipsis">
5.1.4 &nbsp; Typical Applications of Stack
5.1.4 &nbsp; Typical applications of stack
</span>
</a>
@@ -2158,12 +2164,12 @@
<p>A "Stack" is a linear data structure that follows the principle of Last-In-First-Out (LIFO).</p>
<p>We can compare a stack to a pile of plates on a table. To access the bottom plate, one must first remove the plates on top. By replacing the plates with various types of elements (such as integers, characters, objects, etc.), we obtain the data structure known as a stack.</p>
<p>As shown in the Figure 5-1 , we refer to the top of the pile of elements as the "top of the stack" and the bottom as the "bottom of the stack." The operation of adding elements to the top of the stack is called "push," and the operation of removing the top element is called "pop."</p>
<p><a class="glightbox" href="../stack.assets/stack_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Stack's Last-In-First-Out Rule" class="animation-figure" src="../stack.assets/stack_operations.png" /></a></p>
<p align="center"> Figure 5-1 &nbsp; Stack's Last-In-First-Out Rule </p>
<p><a class="glightbox" href="../stack.assets/stack_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Stack's last-in-first-out rule" class="animation-figure" src="../stack.assets/stack_operations.png" /></a></p>
<p align="center"> Figure 5-1 &nbsp; Stack's last-in-first-out rule </p>
<h2 id="511-common-operations-on-stack">5.1.1 &nbsp; Common Operations on Stack<a class="headerlink" href="#511-common-operations-on-stack" title="Permanent link">&para;</a></h2>
<h2 id="511-common-operations-on-stack">5.1.1 &nbsp; Common operations on stack<a class="headerlink" href="#511-common-operations-on-stack" title="Permanent link">&para;</a></h2>
<p>The common operations on a stack are shown in the Table 5-1 . The specific method names depend on the programming language used. Here, we use <code>push()</code>, <code>pop()</code>, and <code>peek()</code> as examples.</p>
<p align="center"> Table 5-1 &nbsp; Efficiency of Stack Operations </p>
<p align="center"> Table 5-1 &nbsp; Efficiency of stack operations </p>
<div class="center-table">
<table>
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<h2 id="512-implementing-a-stack">5.1.2 &nbsp; Implementing a Stack<a class="headerlink" href="#512-implementing-a-stack" title="Permanent link">&para;</a></h2>
<h2 id="512-implementing-a-stack">5.1.2 &nbsp; Implementing a stack<a class="headerlink" href="#512-implementing-a-stack" title="Permanent link">&para;</a></h2>
<p>To gain a deeper understanding of how a stack operates, let's try implementing a stack class ourselves.</p>
<p>A stack follows the principle of Last-In-First-Out, which means we can only add or remove elements at the top of the stack. However, both arrays and linked lists allow adding and removing elements at any position, <strong>therefore a stack can be seen as a restricted array or linked list</strong>. In other words, we can "shield" certain irrelevant operations of an array or linked list, aligning their external behavior with the characteristics of a stack.</p>
<h3 id="1-implementation-based-on-linked-list">1. &nbsp; Implementation Based on Linked List<a class="headerlink" href="#1-implementation-based-on-linked-list" title="Permanent link">&para;</a></h3>
<h3 id="1-implementation-based-on-a-linked-list">1. &nbsp; Implementation based on a linked list<a class="headerlink" href="#1-implementation-based-on-a-linked-list" title="Permanent link">&para;</a></h3>
<p>When implementing a stack using a linked list, we can consider the head node of the list as the top of the stack and the tail node as the bottom of the stack.</p>
<p>As shown in the Figure 5-2 , for the push operation, we simply insert elements at the head of the linked list. This method of node insertion is known as "head insertion." For the pop operation, we just need to remove the head node from the list.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:3"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">LinkedListStack</label><label for="__tabbed_2_2">push()</label><label for="__tabbed_2_3">pop()</label></div>
@@ -3270,7 +3276,7 @@
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</details>
<h3 id="2-implementation-based-on-array">2. &nbsp; Implementation Based on Array<a class="headerlink" href="#2-implementation-based-on-array" title="Permanent link">&para;</a></h3>
<h3 id="2-implementation-based-on-an-array">2. &nbsp; Implementation based on an array<a class="headerlink" href="#2-implementation-based-on-an-array" title="Permanent link">&para;</a></h3>
<p>When implementing a stack using an array, we can consider the end of the array as the top of the stack. As shown in the Figure 5-3 , push and pop operations correspond to adding and removing elements at the end of the array, respectively, both with a time complexity of <span class="arithmatex">\(O(1)\)</span>.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:3"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">ArrayStack</label><label for="__tabbed_4_2">push()</label><label for="__tabbed_4_3">pop()</label></div>
<div class="tabbed-content">
@@ -3907,7 +3913,7 @@
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ArrayStack%3A%0A%20%20%20%20%22%22%22%E5%9F%BA%E4%BA%8E%E6%95%B0%E7%BB%84%E5%AE%9E%E7%8E%B0%E7%9A%84%E6%A0%88%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E6%9E%84%E9%80%A0%E6%96%B9%E6%B3%95%22%22%22%0A%20%20%20%20%20%20%20%20self._stack%3A%20list%5Bint%5D%20%3D%20%5B%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%8E%B7%E5%8F%96%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._stack%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E5%88%A4%E6%96%AD%E6%A0%88%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._stack%20%3D%3D%20%5B%5D%0A%0A%20%20%20%20def%20push%28self,%20item%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E5%85%A5%E6%A0%88%22%22%22%0A%20%20%20%20%20%20%20%20self._stack.append%28item%29%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E5%87%BA%E6%A0%88%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22%E6%A0%88%E4%B8%BA%E7%A9%BA%22%29%0A%20%20%20%20%20%20%20%20return%20self._stack.pop%28%29%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%AE%BF%E9%97%AE%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22%E6%A0%88%E4%B8%BA%E7%A9%BA%22%29%0A%20%20%20%20%20%20%20%20return%20self._stack%5B-1%5D%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%BF%94%E5%9B%9E%E5%88%97%E8%A1%A8%E7%94%A8%E4%BA%8E%E6%89%93%E5%8D%B0%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._stack%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%A0%88%0A%20%20%20%20stack%20%3D%20ArrayStack%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E6%A0%88%0A%20%20%20%20stack.push%281%29%0A%20%20%20%20stack.push%283%29%0A%20%20%20%20stack.push%282%29%0A%20%20%20%20stack.push%285%29%0A%20%20%20%20stack.push%284%29%0A%20%20%20%20print%28%22%E6%A0%88%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20stack.peek%28%29%0A%20%20%20%20print%28%22%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E6%A0%88%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22%E5%87%BA%E6%A0%88%E5%85%83%E7%B4%A0%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22%E5%87%BA%E6%A0%88%E5%90%8E%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20stack.size%28%29%0A%20%20%20%20print%28%22%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20stack.is_empty%28%29%0A%20%20%20%20print%28%22%E6%A0%88%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<h2 id="513-comparison-of-the-two-implementations">5.1.3 &nbsp; Comparison of the Two Implementations<a class="headerlink" href="#513-comparison-of-the-two-implementations" title="Permanent link">&para;</a></h2>
<h2 id="513-comparison-of-the-two-implementations">5.1.3 &nbsp; Comparison of the two implementations<a class="headerlink" href="#513-comparison-of-the-two-implementations" title="Permanent link">&para;</a></h2>
<p><strong>Supported Operations</strong></p>
<p>Both implementations support all the operations defined in a stack. The array implementation additionally supports random access, but this is beyond the scope of a stack definition and is generally not used.</p>
<p><strong>Time Efficiency</strong></p>
@@ -3922,7 +3928,7 @@
<p>When initializing a list, the system allocates an "initial capacity," which might exceed the actual need; moreover, the expansion mechanism usually increases capacity by a specific factor (like doubling), which may also exceed the actual need. Therefore, <strong>the array-based stack might waste some space</strong>.</p>
<p>However, since linked list nodes require extra space for storing pointers, <strong>the space occupied by linked list nodes is relatively larger</strong>.</p>
<p>In summary, we cannot simply determine which implementation is more memory-efficient. It requires analysis based on specific circumstances.</p>
<h2 id="514-typical-applications-of-stack">5.1.4 &nbsp; Typical Applications of Stack<a class="headerlink" href="#514-typical-applications-of-stack" title="Permanent link">&para;</a></h2>
<h2 id="514-typical-applications-of-stack">5.1.4 &nbsp; Typical applications of stack<a class="headerlink" href="#514-typical-applications-of-stack" title="Permanent link">&para;</a></h2>
<ul>
<li><strong>Back and forward in browsers, undo and redo in software</strong>. Every time we open a new webpage, the browser pushes the previous page onto the stack, allowing us to go back to the previous page through the back operation, which is essentially a pop operation. To support both back and forward, two stacks are needed to work together.</li>
<li><strong>Memory management in programs</strong>. Each time a function is called, the system adds a stack frame at the top of the stack to record the function's context information. In recursive functions, the downward recursion phase keeps pushing onto the stack, while the upward backtracking phase keeps popping from the stack.</li>
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<a
href="../"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: Chapter 5. &amp;nbsp; Stack and Queue"
aria-label="Previous: Chapter 5. &amp;nbsp; Stack and queue"
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</span>
<div class="md-ellipsis">
Chapter 5. &nbsp; Stack and Queue
Chapter 5. &nbsp; Stack and queue
</div>
</div>
</a>
@@ -4074,7 +4080,7 @@ aria-label="Footer"
<nav class="md-footer__inner md-grid" aria-label="Footer" >
<a href="../" class="md-footer__link md-footer__link--prev" aria-label="Previous: Chapter 5. &amp;nbsp; Stack and Queue">
<a href="../" class="md-footer__link md-footer__link--prev" aria-label="Previous: Chapter 5. &amp;nbsp; Stack and queue">
<div class="md-footer__button md-icon">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
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</span>
<div class="md-ellipsis">
Chapter 5. &nbsp; Stack and Queue
Chapter 5. &nbsp; Stack and queue
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en/chapter_stack_and_queue/summary/index.html
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
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<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
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3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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3.3 Number Encoding *
3.3 Number encoding *
</span>
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3.4 Character Encoding *
3.4 Character encoding *
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
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<li>Stack is a data structure that follows the Last-In-First-Out (LIFO) principle and can be implemented using arrays or linked lists.</li>
<li>In terms of time efficiency, the array implementation of the stack has a higher average efficiency. However, during expansion, the time complexity for a single push operation can degrade to <span class="arithmatex">\(O(n)\)</span>. In contrast, the linked list implementation of a stack offers more stable efficiency.</li>
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2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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0.2 How to Read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are everywhere
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1.2 What is an algorithm
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and recursion
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2.3 Time complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental data types
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3.3 Number encoding *
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3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked list
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4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1363,7 +1369,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_tree/binary_search_tree/index.html
+37 -31
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@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -996,7 +1002,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1040,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1171,7 +1177,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1363,7 +1369,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_tree/binary_tree/index.html
+37 -31
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@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
@@ -819,7 +825,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
@@ -857,7 +863,7 @@
<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -899,7 +905,7 @@
<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
@@ -996,7 +1002,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1040,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
@@ -1076,7 +1082,7 @@
<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
@@ -1155,7 +1161,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
@@ -1171,7 +1177,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1230,7 +1236,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
@@ -1309,7 +1315,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1325,7 +1331,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1342,7 +1348,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
@@ -1363,7 +1369,7 @@
<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
@@ -1384,7 +1390,7 @@
<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_tree/binary_tree_traversal/index.html
+37 -31
View File
@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
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<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
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<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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6.3 Hash Algorithm
6.3 Hash algorithm
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en/chapter_tree/index.html
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
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<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
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<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
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<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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4.2 Linked List
4.2 Linked list
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4.4 Memory and Cache
4.4 Memory and cache
</span>
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
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5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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<label class="md-nav__title" for="__nav_7">
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Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
en/chapter_tree/summary/index.html
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
@@ -393,7 +399,7 @@
<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
@@ -545,7 +551,7 @@
<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -659,7 +665,7 @@
<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
@@ -680,7 +686,7 @@
<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -803,7 +809,7 @@
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<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -836,7 +842,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -878,7 +884,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
@@ -980,7 +986,7 @@
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<label class="md-nav__title" for="__nav_5">
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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6.1 Hash Table
6.1 Hash table
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6.2 Hash Collision
6.2 Hash collision
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6.3 Hash Algorithm
6.3 Hash algorithm
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en/index.html
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<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
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<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
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0.2 How to Read
0.2 How to read
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
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1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
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1.2 What is an Algorithm
1.2 What is an algorithm
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
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2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
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2.2 Iteration and Recursion
2.2 Iteration and recursion
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2.3 Time Complexity
2.3 Time complexity
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2.4 Space Complexity
2.4 Space complexity
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Chapter 3. Data Structures
Chapter 3. Data structures
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3.1 Classification of Data Structures
3.1 Classification of data structures
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.3 Number Encoding *
3.3 Number encoding *
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3.4 Character Encoding *
3.4 Character encoding *
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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Chapter 4. Array and Linked List
Chapter 4. Array and linked list
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4.2 Linked List
4.2 Linked list
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4.4 Memory and cache
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
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5.3 Double-ended queue
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Chapter 6. Hash Table
Chapter 6. Hash table
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