xinyin025/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2026-07-11 06:56:06 +00:00
Code Issues Packages Projects Releases Wiki Activity

deploy

Browse Source
This commit is contained in:
krahets
2025-12-31 19:38:45 +08:00
parent fdc032e300
commit f31cbc3f9f
474 changed files with 130810 additions and 103691 deletions
Download Patch File Download Diff File Expand all files Collapse all files
en/chapter_computational_complexity/index.html
+96 -96
View File
@@ -37,7 +37,7 @@
<title>Chapter 2.   Complexity analysis - Hello Algo</title>
<title>Chapter 2.   Complexity Analysis - Hello Algo</title>
@@ -58,8 +58,8 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Lato:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Lato";--md-code-font:"JetBrains Mono"}</style>
@@ -154,7 +154,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
Chapter 2. &nbsp; Complexity analysis
Chapter 2. &nbsp; Complexity Analysis
</span>
</div>
@@ -371,7 +371,7 @@
<span class="md-ellipsis">
Before starting
Before Starting
@@ -388,7 +388,7 @@
<span class="md-nav__icon md-icon"></span>
Before starting
Before Starting
</label>
@@ -487,7 +487,7 @@
<span class="md-ellipsis">
0.1 About this book
0.1 About This Book
@@ -515,7 +515,7 @@
<span class="md-ellipsis">
0.2 How to read
0.2 How to Use This Book
@@ -604,7 +604,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
@@ -626,7 +626,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
</label>
@@ -648,7 +648,7 @@
<span class="md-ellipsis">
1.1 Algorithms are everywhere
1.1 Algorithms Are Everywhere
@@ -676,7 +676,7 @@
<span class="md-ellipsis">
1.2 What is an algorithm
1.2 What Is an Algorithm
@@ -771,7 +771,7 @@
<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
@@ -793,7 +793,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
@@ -815,7 +815,7 @@
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
@@ -843,7 +843,7 @@
<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
@@ -871,7 +871,7 @@
<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
@@ -899,7 +899,7 @@
<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
@@ -992,7 +992,7 @@
<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
@@ -1014,7 +1014,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
@@ -1036,7 +1036,7 @@
<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
@@ -1064,7 +1064,7 @@
<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
@@ -1092,7 +1092,7 @@
<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
@@ -1120,7 +1120,7 @@
<span class="md-ellipsis">
3.4 Character encoding *
3.4 Character Encoding *
@@ -1213,7 +1213,7 @@
<span class="md-ellipsis">
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
@@ -1235,7 +1235,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
</label>
@@ -1285,7 +1285,7 @@
<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
@@ -1341,7 +1341,7 @@
<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
@@ -1432,7 +1432,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
@@ -1454,7 +1454,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
</label>
@@ -1532,7 +1532,7 @@
<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
@@ -1623,7 +1623,7 @@
<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
@@ -1645,7 +1645,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
@@ -1667,7 +1667,7 @@
<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
@@ -1695,7 +1695,7 @@
<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
@@ -1723,7 +1723,7 @@
<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
@@ -1862,7 +1862,7 @@
<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
@@ -1890,7 +1890,7 @@
<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
@@ -1918,7 +1918,7 @@
<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
@@ -1946,7 +1946,7 @@
<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
@@ -1974,7 +1974,7 @@
<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
@@ -2137,7 +2137,7 @@
<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
@@ -2165,7 +2165,7 @@
<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
@@ -2328,7 +2328,7 @@
<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
@@ -2356,7 +2356,7 @@
<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
@@ -2495,7 +2495,7 @@
<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
@@ -2523,7 +2523,7 @@
<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
@@ -2551,7 +2551,7 @@
<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
@@ -2579,7 +2579,7 @@
<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
@@ -2607,7 +2607,7 @@
<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
@@ -2756,7 +2756,7 @@
<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
@@ -2784,7 +2784,7 @@
<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
@@ -2812,7 +2812,7 @@
<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
@@ -2840,7 +2840,7 @@
<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
@@ -2868,7 +2868,7 @@
<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
@@ -2896,7 +2896,7 @@
<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
@@ -2924,7 +2924,7 @@
<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
@@ -2952,7 +2952,7 @@
<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
@@ -2980,7 +2980,7 @@
<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
@@ -3008,7 +3008,7 @@
<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
@@ -3101,7 +3101,7 @@
<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
@@ -3123,7 +3123,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
@@ -3145,7 +3145,7 @@
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
@@ -3173,7 +3173,7 @@
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
@@ -3201,7 +3201,7 @@
<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
@@ -3229,7 +3229,7 @@
<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
@@ -3366,7 +3366,7 @@
<span class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
@@ -3394,7 +3394,7 @@
<span class="md-ellipsis">
13.2 Permutation problem
13.2 Permutations Problem
@@ -3422,7 +3422,7 @@
<span class="md-ellipsis">
13.3 Subset sum problem
13.3 Subset-Sum Problem
@@ -3450,7 +3450,7 @@
<span class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
@@ -3547,7 +3547,7 @@
<span class="md-ellipsis">
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
@@ -3569,7 +3569,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
</label>
@@ -3591,7 +3591,7 @@
<span class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
@@ -3619,7 +3619,7 @@
<span class="md-ellipsis">
14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
@@ -3647,7 +3647,7 @@
<span class="md-ellipsis">
14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
@@ -3675,7 +3675,7 @@
<span class="md-ellipsis">
14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
@@ -3703,7 +3703,7 @@
<span class="md-ellipsis">
14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
@@ -3731,7 +3731,7 @@
<span class="md-ellipsis">
14.6 Edit distance problem
14.6 Edit Distance Problem
@@ -3868,7 +3868,7 @@
<span class="md-ellipsis">
15.1 Greedy algorithms
15.1 Greedy Algorithm
@@ -3896,7 +3896,7 @@
<span class="md-ellipsis">
15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
@@ -3924,7 +3924,7 @@
<span class="md-ellipsis">
15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
@@ -3952,7 +3952,7 @@
<span class="md-ellipsis">
15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
@@ -4085,7 +4085,7 @@
<span class="md-ellipsis">
16.1 Installation
16.1 Programming Environment Installation
@@ -4113,7 +4113,7 @@
<span class="md-ellipsis">
16.2 Contributing
16.2 Contributing Together
@@ -4141,7 +4141,7 @@
<span class="md-ellipsis">
16.3 Terminology
16.3 Terminology Table
@@ -4302,19 +4302,19 @@
<!-- 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>
<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>
<p>It guides us in exploring deeper within the dimensions of time and space, seeking more elegant solutions.</p>
<p>Complexity analysis is like a space-time guide in the vast universe of algorithms.</p>
<p>It leads us to explore deeply within the two dimensions of time and space, seeking more elegant solutions.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="performance_evaluation/">2.1 &nbsp; Algorithm efficiency assessment</a></li>
<li><a href="iteration_and_recursion/">2.2 &nbsp; Iteration and recursion</a></li>
<li><a href="time_complexity/">2.3 &nbsp; Time complexity</a></li>
<li><a href="space_complexity/">2.4 &nbsp; Space complexity</a></li>
<li><a href="performance_evaluation/">2.1 &nbsp; Algorithm Efficiency Evaluation</a></li>
<li><a href="iteration_and_recursion/">2.2 &nbsp; Iteration and Recursion</a></li>
<li><a href="time_complexity/">2.3 &nbsp; Time Complexity</a></li>
<li><a href="space_complexity/">2.4 &nbsp; Space Complexity</a></li>
<li><a href="summary/">2.5 &nbsp; Summary</a></li>
</ul>
@@ -4363,7 +4363,7 @@ aria-label="Footer"
<a
href="performance_evaluation/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 2.1 Algorithm efficiency assessment"
aria-label="Next: 2.1 Algorithm Efficiency Evaluation"
rel="next"
>
<div class="md-footer__title">
@@ -4371,7 +4371,7 @@ aria-label="Footer"
Next
</span>
<div class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
</div>
</div>
<div class="md-footer__button md-icon">
@@ -4481,13 +4481,13 @@ aria-label="Footer"
<a href="performance_evaluation/" class="md-footer__link md-footer__link--next" aria-label="Next: 2.1 Algorithm efficiency assessment">
<a href="performance_evaluation/" class="md-footer__link md-footer__link--next" aria-label="Next: 2.1 Algorithm Efficiency Evaluation">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
</div>
</div>
<div class="md-footer__button md-icon">
en/chapter_computational_complexity/iteration_and_recursion/index.html
+1414 -599
View File
File diff suppressed because it is too large Load Diff
en/chapter_computational_complexity/performance_evaluation/index.html
+125 -125
View File
@@ -37,7 +37,7 @@
<title>2.1 Algorithm efficiency assessment - Hello Algo</title>
<title>2.1 Algorithm Efficiency Evaluation - Hello Algo</title>
@@ -58,8 +58,8 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Lato:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Lato";--md-code-font:"JetBrains Mono"}</style>
@@ -99,7 +99,7 @@
<div data-md-component="skip">
<a href="#21-algorithm-efficiency-assessment" class="md-skip">
<a href="#21-algorithm-efficiency-evaluation" class="md-skip">
Skip to content
</a>
@@ -154,7 +154,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
</span>
</div>
@@ -371,7 +371,7 @@
<span class="md-ellipsis">
Before starting
Before Starting
@@ -388,7 +388,7 @@
<span class="md-nav__icon md-icon"></span>
Before starting
Before Starting
</label>
@@ -487,7 +487,7 @@
<span class="md-ellipsis">
0.1 About this book
0.1 About This Book
@@ -515,7 +515,7 @@
<span class="md-ellipsis">
0.2 How to read
0.2 How to Use This Book
@@ -604,7 +604,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
@@ -626,7 +626,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
</label>
@@ -648,7 +648,7 @@
<span class="md-ellipsis">
1.1 Algorithms are everywhere
1.1 Algorithms Are Everywhere
@@ -676,7 +676,7 @@
<span class="md-ellipsis">
1.2 What is an algorithm
1.2 What Is an Algorithm
@@ -771,7 +771,7 @@
<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
@@ -793,7 +793,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
@@ -824,7 +824,7 @@
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
@@ -842,7 +842,7 @@
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
@@ -871,7 +871,7 @@
<a href="#211-actual-testing" class="md-nav__link">
<span class="md-ellipsis">
2.1.1 &nbsp; Actual testing
2.1.1 &nbsp; Actual Testing
</span>
</a>
@@ -882,7 +882,7 @@
<a href="#212-theoretical-estimation" class="md-nav__link">
<span class="md-ellipsis">
2.1.2 &nbsp; Theoretical estimation
2.1.2 &nbsp; Theoretical Estimation
</span>
</a>
@@ -912,7 +912,7 @@
<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
@@ -940,7 +940,7 @@
<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
@@ -968,7 +968,7 @@
<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
@@ -1061,7 +1061,7 @@
<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
@@ -1083,7 +1083,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
@@ -1105,7 +1105,7 @@
<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
@@ -1133,7 +1133,7 @@
<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
@@ -1161,7 +1161,7 @@
<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
@@ -1189,7 +1189,7 @@
<span class="md-ellipsis">
3.4 Character encoding *
3.4 Character Encoding *
@@ -1282,7 +1282,7 @@
<span class="md-ellipsis">
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
@@ -1304,7 +1304,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
</label>
@@ -1354,7 +1354,7 @@
<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
@@ -1410,7 +1410,7 @@
<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
@@ -1501,7 +1501,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
@@ -1523,7 +1523,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
</label>
@@ -1601,7 +1601,7 @@
<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
@@ -1692,7 +1692,7 @@
<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
@@ -1714,7 +1714,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
@@ -1736,7 +1736,7 @@
<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
@@ -1764,7 +1764,7 @@
<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
@@ -1792,7 +1792,7 @@
<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
@@ -1931,7 +1931,7 @@
<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
@@ -1959,7 +1959,7 @@
<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
@@ -1987,7 +1987,7 @@
<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
@@ -2015,7 +2015,7 @@
<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
@@ -2043,7 +2043,7 @@
<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
@@ -2206,7 +2206,7 @@
<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
@@ -2234,7 +2234,7 @@
<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
@@ -2397,7 +2397,7 @@
<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
@@ -2425,7 +2425,7 @@
<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
@@ -2564,7 +2564,7 @@
<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
@@ -2592,7 +2592,7 @@
<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
@@ -2620,7 +2620,7 @@
<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
@@ -2648,7 +2648,7 @@
<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
@@ -2676,7 +2676,7 @@
<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
@@ -2825,7 +2825,7 @@
<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
@@ -2853,7 +2853,7 @@
<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
@@ -2881,7 +2881,7 @@
<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
@@ -2909,7 +2909,7 @@
<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
@@ -2937,7 +2937,7 @@
<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
@@ -2965,7 +2965,7 @@
<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
@@ -2993,7 +2993,7 @@
<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
@@ -3021,7 +3021,7 @@
<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
@@ -3049,7 +3049,7 @@
<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
@@ -3077,7 +3077,7 @@
<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
@@ -3170,7 +3170,7 @@
<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
@@ -3192,7 +3192,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
@@ -3214,7 +3214,7 @@
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
@@ -3242,7 +3242,7 @@
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
@@ -3270,7 +3270,7 @@
<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
@@ -3298,7 +3298,7 @@
<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
@@ -3435,7 +3435,7 @@
<span class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
@@ -3463,7 +3463,7 @@
<span class="md-ellipsis">
13.2 Permutation problem
13.2 Permutations Problem
@@ -3491,7 +3491,7 @@
<span class="md-ellipsis">
13.3 Subset sum problem
13.3 Subset-Sum Problem
@@ -3519,7 +3519,7 @@
<span class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
@@ -3616,7 +3616,7 @@
<span class="md-ellipsis">
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
@@ -3638,7 +3638,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
</label>
@@ -3660,7 +3660,7 @@
<span class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
@@ -3688,7 +3688,7 @@
<span class="md-ellipsis">
14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
@@ -3716,7 +3716,7 @@
<span class="md-ellipsis">
14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
@@ -3744,7 +3744,7 @@
<span class="md-ellipsis">
14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
@@ -3772,7 +3772,7 @@
<span class="md-ellipsis">
14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
@@ -3800,7 +3800,7 @@
<span class="md-ellipsis">
14.6 Edit distance problem
14.6 Edit Distance Problem
@@ -3937,7 +3937,7 @@
<span class="md-ellipsis">
15.1 Greedy algorithms
15.1 Greedy Algorithm
@@ -3965,7 +3965,7 @@
<span class="md-ellipsis">
15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
@@ -3993,7 +3993,7 @@
<span class="md-ellipsis">
15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
@@ -4021,7 +4021,7 @@
<span class="md-ellipsis">
15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
@@ -4154,7 +4154,7 @@
<span class="md-ellipsis">
16.1 Installation
16.1 Programming Environment Installation
@@ -4182,7 +4182,7 @@
<span class="md-ellipsis">
16.2 Contributing
16.2 Contributing Together
@@ -4210,7 +4210,7 @@
<span class="md-ellipsis">
16.3 Terminology
16.3 Terminology Table
@@ -4327,7 +4327,7 @@
<a href="#211-actual-testing" class="md-nav__link">
<span class="md-ellipsis">
2.1.1 &nbsp; Actual testing
2.1.1 &nbsp; Actual Testing
</span>
</a>
@@ -4338,7 +4338,7 @@
<a href="#212-theoretical-estimation" class="md-nav__link">
<span class="md-ellipsis">
2.1.2 &nbsp; Theoretical estimation
2.1.2 &nbsp; Theoretical Estimation
</span>
</a>
@@ -4382,44 +4382,44 @@
<!-- Page content -->
<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>
<h1 id="21-algorithm-efficiency-evaluation">2.1 &nbsp; Algorithm Efficiency Evaluation<a class="headerlink" href="#21-algorithm-efficiency-evaluation" title="Permanent link">&para;</a></h1>
<p>In algorithm design, we pursue the following two levels of objectives sequentially.</p>
<ol>
<li><strong>Finding a Solution to the Problem</strong>: The algorithm should reliably find the correct solution within the specified range of inputs.</li>
<li><strong>Seeking the Optimal Solution</strong>: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.</li>
<li><strong>Finding a solution to the problem</strong>: The algorithm must reliably obtain the correct solution within the specified input range.</li>
<li><strong>Seeking the optimal solution</strong>: Multiple solutions may exist for the same problem, and we hope to find an algorithm that is as efficient as possible.</li>
</ol>
<p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating an algorithm, which includes the following two dimensions.</p>
<p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the primary evaluation criterion for measuring the quality of algorithms. It 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 length of time the algorithm runs.</li>
<li><strong>Space efficiency</strong>: The size of memory space the algorithm occupies.</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>
<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, monitor and record their runtime and memory usage. This assessment method reflects the actual situation, but it 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, an algorithm with a high degree of parallelism is better suited for running on multi-core CPUs, while an algorithm that involves intensive memory operations performs better with high-performance memory. The test results of an algorithm may vary across different machines. This means testing across multiple machines to calculate average efficiency becomes impractical.</p>
<p>On the other hand, <strong>conducting a full test is very resource-intensive</strong>. Algorithm efficiency varies with input data size. For example, with smaller data volumes, algorithm <code>A</code> might run faster than <code>B</code>, but with larger data volumes, the test results may be the opposite. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires excessive 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>
<p>Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as <u>asymptotic complexity analysis</u>, or simply <u>complexity analysis</u>.</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>
<p>In short, <strong>our goal is to design data structures and algorithms that are "both fast and memory-efficient"</strong>. Effectively evaluating algorithm efficiency is crucial, because only in this way can we compare various algorithms and guide the algorithm design and optimization process.</p>
<p>Efficiency evaluation methods are mainly divided into two types: 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>
<p>Suppose we now have algorithm <code>A</code> and algorithm <code>B</code>, both of which can solve the same problem, and we need to compare the efficiency of these two algorithms. The most direct method is to find a computer, run these two algorithms, and monitor and record their running time and memory usage. This evaluation approach can reflect the real situation, but it also has considerable limitations.</p>
<p>On one hand, <strong>it is difficult to eliminate interference factors from the testing environment</strong>. Hardware configuration affects the performance of algorithms. For example, if an algorithm has a high degree of parallelism, it is more suitable for running on multi-core CPUs; if an algorithm has intensive memory operations, it will perform better on high-performance memory. In other words, the test results of an algorithm on different machines may be inconsistent. This means we need to test on various machines and calculate average efficiency, which is impractical.</p>
<p>On the other hand, <strong>conducting complete testing is very resource-intensive</strong>. As the input data volume changes, the algorithm will exhibit different efficiencies. For example, when the input data volume is small, the running time of algorithm <code>A</code> is shorter than algorithm <code>B</code>; but when the input data volume is large, the test results may be exactly the opposite. Therefore, to obtain convincing conclusions, we need to test input data of various scales, which requires a large amount of 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>
<p>Since actual testing has considerable limitations, we can consider evaluating algorithm efficiency through calculations alone. This estimation method is called <u>asymptotic complexity analysis</u>, or <u>complexity analysis</u> for short.</p>
<p>Complexity analysis can reflect the relationship between the time and space resources required for algorithm execution and the input data scale. <strong>It describes the growth trend of the time and space required for algorithm execution as the input data scale increases</strong>. This definition is somewhat convoluted, so we can break it down into three key points to understand.</p>
<ul>
<li>"Time and space resources" correspond to <u>time complexity</u> and <u>space complexity</u>, respectively.</li>
<li>"As the size of input data increases" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.</li>
<li>"The trend of growth in time and space" indicates that complexity analysis focuses not on the specific values of runtime or space occupied, but on the "rate" at which time or space increases.</li>
<li>"As the input data scale increases" means that complexity reflects the relationship between algorithm running efficiency and input data scale.</li>
<li>"Growth trend of time and space" indicates that complexity analysis focuses not on the specific values of running time or occupied space, but on how "fast" time or space grows.</li>
</ul>
<p><strong>Complexity analysis overcomes the disadvantages of actual testing methods</strong>, reflected in the following aspects:</p>
<p><strong>Complexity analysis overcomes the drawbacks of the actual testing method</strong>, reflected in the following aspects.</p>
<ul>
<li>It does not require actually running the code, making it more environmentally friendly and energy efficient.</li>
<li>It is independent of the testing environment and applicable to all operating platforms.</li>
<li>It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.</li>
<li>It does not need to actually run the code, making it more environmentally friendly and energy-efficient.</li>
<li>It is independent of the testing environment, and the analysis results are applicable to all running platforms.</li>
<li>It can reflect algorithm efficiency at different data volumes, especially algorithm performance at large data volumes.</li>
</ul>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>If you're still confused about the concept of complexity, don't worry. We will cover it in detail in subsequent chapters.</p>
<p>If you are still confused about the concept of complexity, don't worry—we will introduce it in detail in subsequent chapters.</p>
</div>
<p>Complexity analysis provides us with a "ruler" to evaluate the efficiency of an algorithm, enabling us to measure the time and space resources required to execute it and compare the efficiency of different algorithms.</p>
<p>Complexity is a mathematical concept that might be abstract and challenging for beginners. From this perspective, complexity analysis might not be the most suitable topic to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.</p>
<p>In summary, it is recommended to develop a basic understanding of complexity analysis before diving deep into data structures and algorithms, <strong>so that you can perform complexity analysis on simple algorithms</strong>.</p>
<p>Complexity analysis provides us with a "ruler" for evaluating algorithm efficiency, allowing us to measure the time and space resources required to execute a certain algorithm and compare the efficiency between different algorithms.</p>
<p>Complexity is a mathematical concept that may be relatively abstract for beginners, with a relatively high learning difficulty. From this perspective, complexity analysis may not be very suitable as the first content to be introduced. However, when we discuss the characteristics of a certain data structure or algorithm, it is difficult to avoid analyzing its running speed and space usage.</p>
<p>In summary, it is recommended that before diving deep into data structures and algorithms, <strong>you first establish a preliminary understanding of complexity analysis so that you can complete complexity analysis of simple algorithms</strong>.</p>
<!-- Source file information -->
@@ -4442,7 +4442,7 @@ aria-label="Footer"
<a
href="../"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: Chapter 2. &amp;nbsp; Complexity analysis"
aria-label="Previous: Chapter 2. &amp;nbsp; Complexity Analysis"
rel="prev"
>
<div class="md-footer__button md-icon">
@@ -4454,7 +4454,7 @@ aria-label="Footer"
Previous
</span>
<div class="md-ellipsis">
Chapter 2. &nbsp; Complexity analysis
Chapter 2. &nbsp; Complexity Analysis
</div>
</div>
</a>
@@ -4466,7 +4466,7 @@ aria-label="Footer"
<a
href="../iteration_and_recursion/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 2.2 Iteration and recursion"
aria-label="Next: 2.2 Iteration and Recursion"
rel="next"
>
<div class="md-footer__title">
@@ -4474,7 +4474,7 @@ aria-label="Footer"
Next
</span>
<div class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
</div>
</div>
<div class="md-footer__button md-icon">
@@ -4567,7 +4567,7 @@ 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 2. &amp;nbsp; Complexity analysis">
<a href="../" class="md-footer__link md-footer__link--prev" aria-label="Previous: Chapter 2. &amp;nbsp; Complexity Analysis">
<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 11z"/></svg>
@@ -4577,20 +4577,20 @@ aria-label="Footer"
Previous
</span>
<div class="md-ellipsis">
Chapter 2. &nbsp; Complexity analysis
Chapter 2. &nbsp; Complexity Analysis
</div>
</div>
</a>
<a href="../iteration_and_recursion/" class="md-footer__link md-footer__link--next" aria-label="Next: 2.2 Iteration and recursion">
<a href="../iteration_and_recursion/" class="md-footer__link md-footer__link--next" aria-label="Next: 2.2 Iteration and Recursion">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
</div>
</div>
<div class="md-footer__button md-icon">
en/chapter_computational_complexity/space_complexity/index.html
+1299 -572
View File
File diff suppressed because it is too large Load Diff
en/chapter_computational_complexity/summary/index.html
+119 -116
View File
@@ -58,8 +58,8 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Lato:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Lato";--md-code-font:"JetBrains Mono"}</style>
@@ -371,7 +371,7 @@
<span class="md-ellipsis">
Before starting
Before Starting
@@ -388,7 +388,7 @@
<span class="md-nav__icon md-icon"></span>
Before starting
Before Starting
</label>
@@ -487,7 +487,7 @@
<span class="md-ellipsis">
0.1 About this book
0.1 About This Book
@@ -515,7 +515,7 @@
<span class="md-ellipsis">
0.2 How to read
0.2 How to Use This Book
@@ -604,7 +604,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
@@ -626,7 +626,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
</label>
@@ -648,7 +648,7 @@
<span class="md-ellipsis">
1.1 Algorithms are everywhere
1.1 Algorithms Are Everywhere
@@ -676,7 +676,7 @@
<span class="md-ellipsis">
1.2 What is an algorithm
1.2 What Is an Algorithm
@@ -771,7 +771,7 @@
<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
@@ -793,7 +793,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
@@ -815,7 +815,7 @@
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
@@ -843,7 +843,7 @@
<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
@@ -871,7 +871,7 @@
<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
@@ -899,7 +899,7 @@
<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
@@ -983,7 +983,7 @@
<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key review
1. &nbsp; Key Review
</span>
</a>
@@ -1061,7 +1061,7 @@
<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
@@ -1083,7 +1083,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
@@ -1105,7 +1105,7 @@
<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
@@ -1133,7 +1133,7 @@
<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
@@ -1161,7 +1161,7 @@
<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
@@ -1189,7 +1189,7 @@
<span class="md-ellipsis">
3.4 Character encoding *
3.4 Character Encoding *
@@ -1282,7 +1282,7 @@
<span class="md-ellipsis">
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
@@ -1304,7 +1304,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
</label>
@@ -1354,7 +1354,7 @@
<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
@@ -1410,7 +1410,7 @@
<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
@@ -1501,7 +1501,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
@@ -1523,7 +1523,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
</label>
@@ -1601,7 +1601,7 @@
<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
@@ -1692,7 +1692,7 @@
<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
@@ -1714,7 +1714,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
@@ -1736,7 +1736,7 @@
<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
@@ -1764,7 +1764,7 @@
<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
@@ -1792,7 +1792,7 @@
<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
@@ -1931,7 +1931,7 @@
<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
@@ -1959,7 +1959,7 @@
<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
@@ -1987,7 +1987,7 @@
<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
@@ -2015,7 +2015,7 @@
<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
@@ -2043,7 +2043,7 @@
<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
@@ -2206,7 +2206,7 @@
<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
@@ -2234,7 +2234,7 @@
<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
@@ -2397,7 +2397,7 @@
<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
@@ -2425,7 +2425,7 @@
<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
@@ -2564,7 +2564,7 @@
<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
@@ -2592,7 +2592,7 @@
<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
@@ -2620,7 +2620,7 @@
<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
@@ -2648,7 +2648,7 @@
<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
@@ -2676,7 +2676,7 @@
<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
@@ -2825,7 +2825,7 @@
<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
@@ -2853,7 +2853,7 @@
<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
@@ -2881,7 +2881,7 @@
<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
@@ -2909,7 +2909,7 @@
<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
@@ -2937,7 +2937,7 @@
<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
@@ -2965,7 +2965,7 @@
<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
@@ -2993,7 +2993,7 @@
<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
@@ -3021,7 +3021,7 @@
<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
@@ -3049,7 +3049,7 @@
<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
@@ -3077,7 +3077,7 @@
<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
@@ -3170,7 +3170,7 @@
<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
@@ -3192,7 +3192,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
@@ -3214,7 +3214,7 @@
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
@@ -3242,7 +3242,7 @@
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
@@ -3270,7 +3270,7 @@
<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
@@ -3298,7 +3298,7 @@
<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
@@ -3435,7 +3435,7 @@
<span class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
@@ -3463,7 +3463,7 @@
<span class="md-ellipsis">
13.2 Permutation problem
13.2 Permutations Problem
@@ -3491,7 +3491,7 @@
<span class="md-ellipsis">
13.3 Subset sum problem
13.3 Subset-Sum Problem
@@ -3519,7 +3519,7 @@
<span class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
@@ -3616,7 +3616,7 @@
<span class="md-ellipsis">
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
@@ -3638,7 +3638,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
</label>
@@ -3660,7 +3660,7 @@
<span class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
@@ -3688,7 +3688,7 @@
<span class="md-ellipsis">
14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
@@ -3716,7 +3716,7 @@
<span class="md-ellipsis">
14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
@@ -3744,7 +3744,7 @@
<span class="md-ellipsis">
14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
@@ -3772,7 +3772,7 @@
<span class="md-ellipsis">
14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
@@ -3800,7 +3800,7 @@
<span class="md-ellipsis">
14.6 Edit distance problem
14.6 Edit Distance Problem
@@ -3937,7 +3937,7 @@
<span class="md-ellipsis">
15.1 Greedy algorithms
15.1 Greedy Algorithm
@@ -3965,7 +3965,7 @@
<span class="md-ellipsis">
15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
@@ -3993,7 +3993,7 @@
<span class="md-ellipsis">
15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
@@ -4021,7 +4021,7 @@
<span class="md-ellipsis">
15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
@@ -4154,7 +4154,7 @@
<span class="md-ellipsis">
16.1 Installation
16.1 Programming Environment Installation
@@ -4182,7 +4182,7 @@
<span class="md-ellipsis">
16.2 Contributing
16.2 Contributing Together
@@ -4210,7 +4210,7 @@
<span class="md-ellipsis">
16.3 Terminology
16.3 Terminology Table
@@ -4327,7 +4327,7 @@
<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key review
1. &nbsp; Key Review
</span>
</a>
@@ -4383,44 +4383,47 @@
<!-- Page content -->
<h1 id="25-summary">2.5 &nbsp; Summary<a class="headerlink" href="#25-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>
<p><strong>Algorithm Efficiency Assessment</strong></p>
<ul>
<li>Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm.</li>
<li>We can assess algorithm efficiency through actual testing, but it's challenging to eliminate the influence of the test environment, and it consumes substantial computational resources.</li>
<li>Complexity analysis can overcome the disadvantages of actual testing. Its results are applicable across all operating platforms and can reveal the efficiency of algorithms at different data scales.</li>
<li>Time efficiency and space efficiency are the two primary evaluation metrics for measuring algorithm performance.</li>
<li>We can evaluate algorithm efficiency through actual testing, but it is difficult to eliminate the influence of the testing environment, and it consumes substantial computational resources.</li>
<li>Complexity analysis can eliminate the drawbacks of actual testing, with results applicable to all running platforms, and it can reveal algorithm efficiency under different data scales.</li>
</ul>
<p><strong>Time Complexity</strong></p>
<ul>
<li>Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms.</li>
<li>Worst-case time complexity is denoted using big-<span class="arithmatex">\(O\)</span> notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> approaches infinity.</li>
<li>Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound.</li>
<li>Common time complexities, arranged from low to high, include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n \log n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, <span class="arithmatex">\(O(2^n)\)</span>, and <span class="arithmatex">\(O(n!)\)</span>, among others.</li>
<li>The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case.</li>
<li>Average time complexity reflects the efficiency of an algorithm under random data inputs, closely resembling the algorithm's performance in actual applications. Calculating average time complexity requires accounting for the distribution of input data and the subsequent mathematical expectation.</li>
<li>Time complexity is used to measure the trend of algorithm runtime as data volume increases. It can effectively evaluate algorithm efficiency, but may fail in certain situations, such as when the input data volume is small or when time complexities are identical, making it impossible to precisely compare algorithm efficiency.</li>
<li>Worst-case time complexity is represented using Big <span class="arithmatex">\(O\)</span> notation, corresponding to the asymptotic upper bound of a function, reflecting the growth level of the number of operations <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> approaches positive infinity.</li>
<li>Deriving time complexity involves two steps: first, counting the number of operations, then determining the asymptotic upper bound.</li>
<li>Common time complexities arranged from low to high include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n \log n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, <span class="arithmatex">\(O(2^n)\)</span>, and <span class="arithmatex">\(O(n!)\)</span>.</li>
<li>The time complexity of some algorithms is not fixed, but rather depends on the distribution of input data. Time complexity is divided into worst-case, best-case, and average-case time complexity. Best-case time complexity is rarely used because input data generally needs to satisfy strict conditions to achieve the best case.</li>
<li>Average time complexity reflects the algorithm's runtime efficiency under random data input, and is closest to the algorithm's performance in practical applications. Calculating average time complexity requires statistical analysis of input data distribution and the combined mathematical expectation.</li>
</ul>
<p><strong>Space Complexity</strong></p>
<ul>
<li>Space complexity, similar to time complexity, measures the trend of memory space occupied by an algorithm with the increase in data volume.</li>
<li>The relevant memory space used during the algorithm's execution can be divided into input space, temporary space, and output space. Generally, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.</li>
<li>We usually focus only on the worst-case space complexity, which means calculating the space complexity of the algorithm under the worst input data and at the worst moment of operation.</li>
<li>Common space complexities, arranged from low to high, include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, and <span class="arithmatex">\(O(2^n)\)</span>, among others.</li>
<li>Space complexity serves a similar purpose to time complexity, used to measure the trend of algorithm memory usage as data volume increases.</li>
<li>The memory space related to algorithm execution can be divided into input space, temporary space, and output space. Typically, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.</li>
<li>We typically only focus on worst-case space complexity, which is the space complexity of an algorithm under worst-case input data and worst-case runtime.</li>
<li>Common space complexities arranged from low to high include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, and <span class="arithmatex">\(O(2^n)\)</span>.</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>: Is the space complexity of tail recursion <span class="arithmatex">\(O(1)\)</span>?</p>
<p>Theoretically, the space complexity of a tail-recursive function can be optimized to <span class="arithmatex">\(O(1)\)</span>. However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
<p><strong>Q</strong>: What is the difference between the terms "function" and "method"?</p>
<p>A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.</p>
<p>Here are some examples from common programming languages:</p>
<p>Theoretically, the space complexity of tail recursive functions can be optimized to <span class="arithmatex">\(O(1)\)</span>. However, most programming languages (such as Java, Python, C++, Go, C#, etc.) do not support automatic tail recursion optimization, so the space complexity is generally considered to be <span class="arithmatex">\(O(n)\)</span>.</p>
<p><strong>Q</strong>: What is the difference between the terms function and method?</p>
<p>A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object, is implicitly passed to the object that invokes it, and can operate on data contained in class instances.</p>
<p>The following examples use several common programming languages for illustration.</p>
<ul>
<li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages.</li>
<li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with structures are equivalent to methods in other programming languages.</li>
<li>Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.</li>
<li>C++ and Python support both procedural programming (functions) and object-oriented programming (methods).</li>
</ul>
<p><strong>Q</strong>: Does the "Common Types of Space Complexity" figure reflect the absolute size of occupied space?</p>
<p>No, the figure shows space complexities, which reflect growth trends, not the absolute size of the occupied space.</p>
<p>If you take <span class="arithmatex">\(n = 8\)</span>, you might find that the values of each curve don't correspond to their functions. This is because each curve includes a constant term, intended to compress the value range into a visually comfortable range.</p>
<p>In practice, since we usually don't know the "constant term" complexity of each method, it's generally not possible to choose the best solution for <span class="arithmatex">\(n = 8\)</span> based solely on complexity. However, for <span class="arithmatex">\(n = 8^5\)</span>, it's much easier to choose, as the growth trend becomes dominant.</p>
<p><strong>Q</strong>: Does the diagram for "common space complexity types" reflect the absolute size of occupied space?</p>
<p>No, the diagram shows space complexity, which reflects growth trends rather than the absolute size of occupied space.</p>
<p>Assuming <span class="arithmatex">\(n = 8\)</span>, you might find that the values of each curve do not correspond to the functions. This is because each curve contains a constant term used to compress the value range into a visually comfortable range.</p>
<p>In practice, because we generally do not know what the "constant term" complexity of each method is, we usually cannot select the optimal solution for <span class="arithmatex">\(n = 8\)</span> based on complexity alone. But for <span class="arithmatex">\(n = 8^5\)</span>, the choice is straightforward, as the growth trend already dominates.</p>
<p><strong>Q</strong>: Are there situations where algorithms are designed to sacrifice time (or space) based on actual use cases?</p>
<p>In practical applications, most situations choose to sacrifice space for time. For example, with database indexes, we typically choose to build B+ trees or hash indexes, occupying substantial memory space in exchange for efficient queries of <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>.</p>
<p>In scenarios where space resources are precious, time may be sacrificed for space. For example, in embedded development, device memory is precious, and engineers may forgo using hash tables and choose to use array sequential search to save memory usage, at the cost of slower searches.</p>
<!-- Source file information -->
@@ -4443,7 +4446,7 @@ aria-label="Footer"
<a
href="../space_complexity/"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: 2.4 Space complexity"
aria-label="Previous: 2.4 Space Complexity"
rel="prev"
>
<div class="md-footer__button md-icon">
@@ -4455,7 +4458,7 @@ aria-label="Footer"
Previous
</span>
<div class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
</div>
</div>
</a>
@@ -4467,7 +4470,7 @@ aria-label="Footer"
<a
href="../../chapter_data_structure/"
class="md-footer__link md-footer__link--next"
aria-label="Next: Chapter 3. &amp;nbsp; Data structures"
aria-label="Next: Chapter 3. &amp;nbsp; Data Structures"
rel="next"
>
<div class="md-footer__title">
@@ -4475,7 +4478,7 @@ aria-label="Footer"
Next
</span>
<div class="md-ellipsis">
Chapter 3. &nbsp; Data structures
Chapter 3. &nbsp; Data Structures
</div>
</div>
<div class="md-footer__button md-icon">
@@ -4568,7 +4571,7 @@ aria-label="Footer"
<nav class="md-footer__inner md-grid" aria-label="Footer" >
<a href="../space_complexity/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 2.4 Space complexity">
<a href="../space_complexity/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 2.4 Space Complexity">
<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 11z"/></svg>
@@ -4578,20 +4581,20 @@ aria-label="Footer"
Previous
</span>
<div class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
</div>
</div>
</a>
<a href="../../chapter_data_structure/" class="md-footer__link md-footer__link--next" aria-label="Next: Chapter 3. &amp;nbsp; Data structures">
<a href="../../chapter_data_structure/" class="md-footer__link md-footer__link--next" aria-label="Next: Chapter 3. &amp;nbsp; Data Structures">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
Chapter 3. &nbsp; Data structures
Chapter 3. &nbsp; Data Structures
</div>
</div>
<div class="md-footer__button md-icon">
en/chapter_computational_complexity/time_complexity/index.html
+2187 -1001
View File
File diff suppressed because it is too large Load Diff