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<title>5.3 Double-ended Queue - Hello Algo</title>
<title>5.3 Double-ended queue - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended 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>
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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>
@@ -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">
<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>
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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">
<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>
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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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<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>
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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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<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>
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<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>
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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="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>
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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">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<li class="md-nav__item">
<a href="#531-common-operations-in-double-ended-queue" class="md-nav__link">
<span class="md-ellipsis">
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">
<span class="md-ellipsis">
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">
<span class="md-ellipsis">
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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<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
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<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>
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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="#531-common-operations-in-double-ended-queue" class="md-nav__link">
<span class="md-ellipsis">
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">
<span class="md-ellipsis">
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">
<span class="md-ellipsis">
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>
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<title>Chapter 5.   Stack and Queue - Hello Algo</title>
<title>Chapter 5.   Stack and queue - Hello Algo</title>
@@ -153,7 +153,7 @@
<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>
@@ -507,7 +513,7 @@
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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
</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 @@
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<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>
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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>
@@ -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 @@
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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 @@
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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>
@@ -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 @@
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<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>
@@ -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 @@
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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>
@@ -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-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">
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Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
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@@ -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>
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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>
@@ -819,7 +825,7 @@
<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>
<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>
@@ -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="true">
<label class="md-nav__title" for="__nav_6">
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Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
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@@ -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">
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<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">
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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>
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<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>
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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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<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
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<li class="md-nav__item">
<a href="#511-common-operations-on-stack" class="md-nav__link">
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5.1.1 &nbsp; Common Operations on Stack
5.1.1 &nbsp; Common operations on stack
</span>
</a>
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<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">
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<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">
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2. &nbsp; Implementation Based on Array
2. &nbsp; Implementation based on an array
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<a href="#513-comparison-of-the-two-implementations" class="md-nav__link">
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5.1.3 &nbsp; Comparison of the Two Implementations
5.1.3 &nbsp; Comparison of the two implementations
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</a>
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<a href="#514-typical-applications-of-stack" class="md-nav__link">
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5.1.4 &nbsp; Typical Applications of Stack
5.1.4 &nbsp; Typical applications of stack
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<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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</details>
<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
</span>
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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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@@ -1232,7 +1238,7 @@
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5.3 Double-ended Queue
5.3 Double-ended queue
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1. &nbsp; Key Review
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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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6.3 Hash Algorithm
6.3 Hash algorithm
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<h1 id="54-summary">5.4 &nbsp; Summary<a class="headerlink" href="#54-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>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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