xinyin025/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2026-07-15 08:26:06 +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_backtracking/backtracking_algorithm/index.html
+1576 -761
View File
File diff suppressed because it is too large Load Diff
en/chapter_backtracking/index.html
+93 -93
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
@@ -769,7 +769,7 @@
<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
@@ -791,7 +791,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
@@ -813,7 +813,7 @@
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
@@ -841,7 +841,7 @@
<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
@@ -869,7 +869,7 @@
<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
@@ -897,7 +897,7 @@
<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
@@ -990,7 +990,7 @@
<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
@@ -1012,7 +1012,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
@@ -1034,7 +1034,7 @@
<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
@@ -1062,7 +1062,7 @@
<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
@@ -1090,7 +1090,7 @@
<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
@@ -1118,7 +1118,7 @@
<span class="md-ellipsis">
3.4 Character encoding *
3.4 Character Encoding *
@@ -1211,7 +1211,7 @@
<span class="md-ellipsis">
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
@@ -1233,7 +1233,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
</label>
@@ -1283,7 +1283,7 @@
<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
@@ -1339,7 +1339,7 @@
<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
@@ -1430,7 +1430,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
@@ -1452,7 +1452,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
</label>
@@ -1530,7 +1530,7 @@
<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
@@ -1621,7 +1621,7 @@
<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
@@ -1643,7 +1643,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
@@ -1665,7 +1665,7 @@
<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
@@ -1693,7 +1693,7 @@
<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
@@ -1721,7 +1721,7 @@
<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
@@ -1860,7 +1860,7 @@
<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
@@ -1888,7 +1888,7 @@
<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
@@ -1916,7 +1916,7 @@
<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
@@ -1944,7 +1944,7 @@
<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
@@ -1972,7 +1972,7 @@
<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
@@ -2135,7 +2135,7 @@
<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
@@ -2163,7 +2163,7 @@
<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
@@ -2326,7 +2326,7 @@
<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
@@ -2354,7 +2354,7 @@
<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
@@ -2493,7 +2493,7 @@
<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
@@ -2521,7 +2521,7 @@
<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
@@ -2549,7 +2549,7 @@
<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
@@ -2577,7 +2577,7 @@
<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
@@ -2605,7 +2605,7 @@
<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
@@ -2754,7 +2754,7 @@
<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
@@ -2782,7 +2782,7 @@
<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
@@ -2810,7 +2810,7 @@
<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
@@ -2838,7 +2838,7 @@
<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
@@ -2866,7 +2866,7 @@
<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
@@ -2894,7 +2894,7 @@
<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
@@ -2922,7 +2922,7 @@
<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
@@ -2950,7 +2950,7 @@
<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
@@ -2978,7 +2978,7 @@
<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
@@ -3006,7 +3006,7 @@
<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
@@ -3099,7 +3099,7 @@
<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
@@ -3121,7 +3121,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
@@ -3143,7 +3143,7 @@
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
@@ -3171,7 +3171,7 @@
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
@@ -3199,7 +3199,7 @@
<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
@@ -3227,7 +3227,7 @@
<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
@@ -4306,15 +4306,15 @@
<p><a class="glightbox" href="../assets/covers/chapter_backtracking.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Backtracking" class="cover-image" src="../assets/covers/chapter_backtracking.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Like explorers in a maze, we may encounter obstacles on our path forward.</p>
<p>The power of backtracking lets us begin anew, keep trying, and eventually find the exit leading to the light.</p>
<p>We are like explorers in a maze, and may encounter difficulties on the path forward.</p>
<p>The power of backtracking allows us to start over, keep trying, and eventually find the exit leading to light.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="backtracking_algorithm/">13.1 &nbsp; Backtracking algorithms</a></li>
<li><a href="permutations_problem/">13.2 &nbsp; Permutation problem</a></li>
<li><a href="subset_sum_problem/">13.3 &nbsp; Subset sum problem</a></li>
<li><a href="n_queens_problem/">13.4 &nbsp; n queens problem</a></li>
<li><a href="backtracking_algorithm/">13.1 &nbsp; Backtracking Algorithm</a></li>
<li><a href="permutations_problem/">13.2 &nbsp; Permutations Problem</a></li>
<li><a href="subset_sum_problem/">13.3 &nbsp; Subset-Sum Problem</a></li>
<li><a href="n_queens_problem/">13.4 &nbsp; N-Queens Problem</a></li>
<li><a href="summary/">13.5 &nbsp; Summary</a></li>
</ul>
@@ -4363,7 +4363,7 @@ aria-label="Footer"
<a
href="backtracking_algorithm/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 13.1 Backtracking algorithms"
aria-label="Next: 13.1 Backtracking Algorithm"
rel="next"
>
<div class="md-footer__title">
@@ -4371,7 +4371,7 @@ aria-label="Footer"
Next
</span>
<div class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
</div>
</div>
<div class="md-footer__button md-icon">
@@ -4481,13 +4481,13 @@ aria-label="Footer"
<a href="backtracking_algorithm/" class="md-footer__link md-footer__link--next" aria-label="Next: 13.1 Backtracking algorithms">
<a href="backtracking_algorithm/" class="md-footer__link md-footer__link--next" aria-label="Next: 13.1 Backtracking Algorithm">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
</div>
</div>
<div class="md-footer__button md-icon">
en/chapter_backtracking/n_queens_problem/index.html
+645 -190
View File
File diff suppressed because it is too large Load Diff
en/chapter_backtracking/permutations_problem/index.html
+936 -327
View File
File diff suppressed because it is too large Load Diff
en/chapter_backtracking/subset_sum_problem/index.html
+1433 -460
View File
File diff suppressed because it is too large Load Diff
en/chapter_backtracking/summary/index.html
+108 -108
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
@@ -769,7 +769,7 @@
<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
@@ -791,7 +791,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
@@ -813,7 +813,7 @@
<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
@@ -841,7 +841,7 @@
<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
@@ -869,7 +869,7 @@
<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
@@ -897,7 +897,7 @@
<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
@@ -990,7 +990,7 @@
<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
@@ -1012,7 +1012,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
@@ -1034,7 +1034,7 @@
<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
@@ -1062,7 +1062,7 @@
<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
@@ -1090,7 +1090,7 @@
<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
@@ -1118,7 +1118,7 @@
<span class="md-ellipsis">
3.4 Character encoding *
3.4 Character Encoding *
@@ -1211,7 +1211,7 @@
<span class="md-ellipsis">
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
@@ -1233,7 +1233,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
</label>
@@ -1283,7 +1283,7 @@
<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
@@ -1339,7 +1339,7 @@
<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
@@ -1430,7 +1430,7 @@
<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
@@ -1452,7 +1452,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
</label>
@@ -1530,7 +1530,7 @@
<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
@@ -1621,7 +1621,7 @@
<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
@@ -1643,7 +1643,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
@@ -1665,7 +1665,7 @@
<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
@@ -1693,7 +1693,7 @@
<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
@@ -1721,7 +1721,7 @@
<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
@@ -1860,7 +1860,7 @@
<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
@@ -1888,7 +1888,7 @@
<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
@@ -1916,7 +1916,7 @@
<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
@@ -1944,7 +1944,7 @@
<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
@@ -1972,7 +1972,7 @@
<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
@@ -2135,7 +2135,7 @@
<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
@@ -2163,7 +2163,7 @@
<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
@@ -2326,7 +2326,7 @@
<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
@@ -2354,7 +2354,7 @@
<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
@@ -2493,7 +2493,7 @@
<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
@@ -2521,7 +2521,7 @@
<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
@@ -2549,7 +2549,7 @@
<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
@@ -2577,7 +2577,7 @@
<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
@@ -2605,7 +2605,7 @@
<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
@@ -2754,7 +2754,7 @@
<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
@@ -2782,7 +2782,7 @@
<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
@@ -2810,7 +2810,7 @@
<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
@@ -2838,7 +2838,7 @@
<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
@@ -2866,7 +2866,7 @@
<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
@@ -2894,7 +2894,7 @@
<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
@@ -2922,7 +2922,7 @@
<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
@@ -2950,7 +2950,7 @@
<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
@@ -2978,7 +2978,7 @@
<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
@@ -3006,7 +3006,7 @@
<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
@@ -3099,7 +3099,7 @@
<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
@@ -3121,7 +3121,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
@@ -3143,7 +3143,7 @@
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
@@ -3171,7 +3171,7 @@
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
@@ -3199,7 +3199,7 @@
<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
@@ -3227,7 +3227,7 @@
<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
@@ -3534,7 +3534,7 @@
<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key review
1. &nbsp; Key Review
</span>
</a>
@@ -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>
@@ -4383,25 +4383,25 @@
<!-- Page content -->
<h1 id="135-summary">13.5 &nbsp; Summary<a class="headerlink" href="#135-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>The essence of the backtracking algorithm is exhaustive search. It seeks solutions that meet the conditions by performing a depth-first traversal of the solution space. During the search, if a satisfying solution is found, it is recorded, until all solutions are found or the traversal is completed.</li>
<li>The search process of the backtracking algorithm includes trying and backtracking. It uses depth-first search to explore various choices, and when a choice does not meet the constraints, the previous choice is undone. Then it reverts to the previous state and continues to try other options. Trying and backtracking are operations in opposite directions.</li>
<li>Backtracking problems usually contain multiple constraints. These constraints can be used to perform pruning operations. Pruning can terminate unnecessary search branches in advance, greatly enhancing search efficiency.</li>
<li>The backtracking algorithm is mainly used to solve search problems and constraint satisfaction problems. Although combinatorial optimization problems can be solved using backtracking, there are often more efficient or effective solutions available.</li>
<li>The permutation problem aims to search for all possible permutations of the elements in a given set. We use an array to record whether each element has been chosen, avoiding repeated selection of the same element. This ensures that each element is chosen only once.</li>
<li>In permutation problems, if the set contains duplicate elements, the final result will include duplicate permutations. We need to restrict that identical elements can only be selected once in each round, which is usually implemented using a hash set.</li>
<li>The subset-sum problem aims to find all subsets in a given set that sum to a target value. The set does not distinguish the order of elements, but the search process may generate duplicate subsets. This occurs because the algorithm explores different element orders as unique paths. Before backtracking, we sort the data and set a variable to indicate the starting point of the traversal for each round. This allows us to prune the search branches that generate duplicate subsets.</li>
<li>For the subset-sum problem, equal elements in the array can produce duplicate sets. Using the precondition that the array is already sorted, we prune by determining if adjacent elements are equal. This ensures that equal elements are only selected once per round.</li>
<li>The <span class="arithmatex">\(n\)</span> queens problem aims to find schemes to place <span class="arithmatex">\(n\)</span> queens on an <span class="arithmatex">\(n \times n\)</span> chessboard such that no two queens can attack each other. The constraints of the problem include row constraints, column constraints, and constraints on the main and secondary diagonals. To meet the row constraint, we adopt a strategy of placing one queen per row, ensuring each row has one queen placed.</li>
<li>The handling of column constraints and diagonal constraints is similar. For column constraints, we use an array to record whether there is a queen in each column, thereby indicating whether the selected cell is legal. For diagonal constraints, we use two arrays to respectively record the presence of queens on the main and secondary diagonals. The challenge is to determine the relationship between row and column indices for cells on the same main or secondary diagonal.</li>
<li>The backtracking algorithm is fundamentally an exhaustive search method. It finds solutions that meet specified conditions by performing a depth-first traversal of the solution space. During the search process, when a solution satisfying the conditions is found, it is recorded. The search ends either after finding all solutions or when the traversal is complete.</li>
<li>The backtracking algorithm search process consists of two parts: attempting and backtracking. It tries various choices through depth-first search. When encountering situations that violate constraints, it reverts the previous choice, returns to the previous state, and continues exploring other options. Attempting and backtracking are operations in opposite directions.</li>
<li>Backtracking problems typically contain multiple constraints, which can be utilized to implement pruning operations. Pruning can terminate unnecessary search branches early, significantly improving search efficiency.</li>
<li>The backtracking algorithm is primarily used to solve search problems and constraint satisfaction problems. While combinatorial optimization problems can be solved with backtracking, there are often more efficient or better-performing solutions available.</li>
<li>The permutation problem aims to find all possible permutations of elements in a given set. We use an array to record whether each element has been selected, thereby pruning search branches that attempt to select the same element repeatedly, ensuring each element is selected exactly once.</li>
<li>In the permutation problem, if the set contains duplicate elements, the final result will contain duplicate permutations. We need to impose a constraint so that equal elements can only be selected once per round, which is typically achieved using a hash set.</li>
<li>The subset-sum problem aims to find all subsets of a given set that sum to a target value. Since the set is unordered but the search process outputs results in all orders, duplicate subsets are generated. We sort the data before backtracking and use a variable to indicate the starting point of each round's traversal, thereby pruning search branches that generate duplicate subsets.</li>
<li>For the subset-sum problem, equal elements in the array produce duplicate sets. We leverage the precondition that the array is sorted by checking whether adjacent elements are equal to implement pruning, ensuring that equal elements can only be selected once per round.</li>
<li>The <span class="arithmatex">\(n\)</span> queens problem aims to find placements of <span class="arithmatex">\(n\)</span> queens on an <span class="arithmatex">\(n \times n\)</span> chessboard such that no two queens can attack each other. The constraints of this problem include row constraints, column constraints, and main and anti-diagonal constraints. To satisfy row constraints, we adopt a row-by-row placement strategy, ensuring exactly one queen is placed in each row.</li>
<li>The handling of column constraints and diagonal constraints is similar. For column constraints, we use an array to record whether each column has a queen, thereby indicating whether a selected cell is valid. For diagonal constraints, we use two arrays to separately record whether queens exist on each main or anti-diagonal. The challenge lies in finding the row-column index pattern that characterizes cells on the same main (anti-)diagonal.</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>: How can we understand the relationship between backtracking and recursion?</p>
<p>Overall, backtracking is an "algorithmic strategy," while recursion is more of a "tool."</p>
<p><strong>Q</strong>: How should we understand the relationship between backtracking and recursion?</p>
<p>Overall, backtracking is an "algorithm strategy", while recursion is more like a "tool".</p>
<ul>
<li>Backtracking algorithms are typically based on recursion. However, backtracking is one of the application scenarios of recursion, specifically in search problems.</li>
<li>The structure of recursion reflects the problem-solving paradigm of "sub-problem decomposition." It is commonly used in solving problems involving divide and conquer, backtracking, and dynamic programming (memoized recursion).</li>
<li>The backtracking algorithm is typically implemented based on recursion. However, backtracking is one application scenario of recursion and represents the application of recursion in search problems.</li>
<li>The structure of recursion embodies the "subproblem decomposition" problem-solving paradigm, commonly used to solve problems involving divide-and-conquer, backtracking, and dynamic programming (memoized recursion).</li>
</ul>
<!-- Source file information -->
@@ -4425,7 +4425,7 @@ aria-label="Footer"
<a
href="../n_queens_problem/"
class="md-footer__link md-footer__link--prev"
aria-label="Previous: 13.4 n queens problem"
aria-label="Previous: 13.4 N-Queens Problem"
rel="prev"
>
<div class="md-footer__button md-icon">
@@ -4437,7 +4437,7 @@ aria-label="Footer"
Previous
</span>
<div class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
</div>
</div>
</a>
@@ -4449,7 +4449,7 @@ aria-label="Footer"
<a
href="../../chapter_dynamic_programming/"
class="md-footer__link md-footer__link--next"
aria-label="Next: Chapter 14. &amp;nbsp; Dynamic programming"
aria-label="Next: Chapter 14. &amp;nbsp; Dynamic Programming"
rel="next"
>
<div class="md-footer__title">
@@ -4457,7 +4457,7 @@ aria-label="Footer"
Next
</span>
<div class="md-ellipsis">
Chapter 14. &nbsp; Dynamic programming
Chapter 14. &nbsp; Dynamic Programming
</div>
</div>
<div class="md-footer__button md-icon">
@@ -4550,7 +4550,7 @@ aria-label="Footer"
<nav class="md-footer__inner md-grid" aria-label="Footer" >
<a href="../n_queens_problem/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 13.4 n queens problem">
<a href="../n_queens_problem/" class="md-footer__link md-footer__link--prev" aria-label="Previous: 13.4 N-Queens Problem">
<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>
@@ -4560,20 +4560,20 @@ aria-label="Footer"
Previous
</span>
<div class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
</div>
</div>
</a>
<a href="../../chapter_dynamic_programming/" class="md-footer__link md-footer__link--next" aria-label="Next: Chapter 14. &amp;nbsp; Dynamic programming">
<a href="../../chapter_dynamic_programming/" class="md-footer__link md-footer__link--next" aria-label="Next: Chapter 14. &amp;nbsp; Dynamic Programming">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
Chapter 14. &nbsp; Dynamic programming
Chapter 14. &nbsp; Dynamic Programming
</div>
</div>
<div class="md-footer__button md-icon">