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<title>Chapter 14.   Dynamic programming - Hello Algo</title>
<title>Chapter 14.   Dynamic Programming - Hello Algo</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
Chapter 14. &nbsp; Dynamic programming
Chapter 14. &nbsp; Dynamic Programming
</span>
</div>
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<span class="md-ellipsis">
Before starting
Before Starting
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<span class="md-nav__icon md-icon"></span>
Before starting
Before Starting
</label>
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<span class="md-ellipsis">
0.1 About this book
0.1 About This Book
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<span class="md-ellipsis">
0.2 How to read
0.2 How to Use This Book
@@ -604,7 +604,7 @@
<span class="md-ellipsis">
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
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<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
</label>
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<span class="md-ellipsis">
1.1 Algorithms are everywhere
1.1 Algorithms Are Everywhere
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<span class="md-ellipsis">
1.2 What is an algorithm
1.2 What Is an Algorithm
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<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
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<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
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<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
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<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
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<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
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<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
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<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
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<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
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<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
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<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
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<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
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<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
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<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
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<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
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<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
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<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
</label>
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<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
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<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
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<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
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<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
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<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
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<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
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<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
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<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
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<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
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<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
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<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
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<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
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<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
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<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
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<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
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<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
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<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
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<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
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<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
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<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
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<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
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<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
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<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
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<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
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<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
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<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
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<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
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<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
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<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
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<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
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Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
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<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
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<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
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<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
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<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
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<span class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
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<span class="md-ellipsis">
13.2 Permutation problem
13.2 Permutations Problem
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<span class="md-ellipsis">
13.3 Subset sum problem
13.3 Subset-Sum Problem
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<span class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
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<span class="md-ellipsis">
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
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<span class="md-nav__icon md-icon"></span>
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
</label>
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<span class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
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<span class="md-ellipsis">
14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
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<span class="md-ellipsis">
14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
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<span class="md-ellipsis">
14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
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<span class="md-ellipsis">
14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
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<span class="md-ellipsis">
14.6 Edit distance problem
14.6 Edit Distance Problem
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<span class="md-ellipsis">
15.1 Greedy algorithms
15.1 Greedy Algorithm
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<span class="md-ellipsis">
15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
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<span class="md-ellipsis">
15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
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<span class="md-ellipsis">
15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
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<span class="md-ellipsis">
16.1 Installation
16.1 Programming Environment Installation
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<span class="md-ellipsis">
16.2 Contributing
16.2 Contributing Together
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<span class="md-ellipsis">
16.3 Terminology
16.3 Terminology Table
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<!-- Page content -->
<h1 id="chapter-14-dynamic-programming">Chapter 14. &nbsp; Dynamic programming<a class="headerlink" href="#chapter-14-dynamic-programming" title="Permanent link">&para;</a></h1>
<h1 id="chapter-14-dynamic-programming">Chapter 14. &nbsp; Dynamic Programming<a class="headerlink" href="#chapter-14-dynamic-programming" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_dynamic_programming.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Dynamic programming" class="cover-image" src="../assets/covers/chapter_dynamic_programming.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Streams merge into rivers, and rivers merge into the sea.</p>
<p>Dynamic programming weaves smaller problems’ solutions into larger ones, guiding us step by step toward the far shore—where the ultimate answer awaits.</p>
<p>Streams converge into rivers, rivers converge into the sea.</p>
<p>Dynamic programming gathers solutions to small problems into answers to large problems, step by step guiding us to the shore of problem-solving.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="intro_to_dynamic_programming/">14.1 &nbsp; Introduction to dynamic programming</a></li>
<li><a href="dp_problem_features/">14.2 &nbsp; Characteristics of DP problems</a></li>
<li><a href="dp_solution_pipeline/">14.3 &nbsp; DP problem-solving approach</a></li>
<li><a href="knapsack_problem/">14.4 &nbsp; 0-1 Knapsack problem</a></li>
<li><a href="unbounded_knapsack_problem/">14.5 &nbsp; Unbounded knapsack problem</a></li>
<li><a href="edit_distance_problem/">14.6 &nbsp; Edit distance problem</a></li>
<li><a href="intro_to_dynamic_programming/">14.1 &nbsp; Introduction to Dynamic Programming</a></li>
<li><a href="dp_problem_features/">14.2 &nbsp; Characteristics of Dynamic Programming Problems</a></li>
<li><a href="dp_solution_pipeline/">14.3 &nbsp; Dynamic Programming Problem-Solving Approach</a></li>
<li><a href="knapsack_problem/">14.4 &nbsp; 0-1 Knapsack Problem</a></li>
<li><a href="unbounded_knapsack_problem/">14.5 &nbsp; Unbounded Knapsack Problem</a></li>
<li><a href="edit_distance_problem/">14.6 &nbsp; Edit Distance Problem</a></li>
<li><a href="summary/">14.7 &nbsp; Summary</a></li>
</ul>
@@ -4365,7 +4365,7 @@ aria-label="Footer"
<a
href="intro_to_dynamic_programming/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 14.1 Introduction to dynamic programming"
aria-label="Next: 14.1 Introduction to Dynamic Programming"
rel="next"
>
<div class="md-footer__title">
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Next
</span>
<div class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
</div>
</div>
<div class="md-footer__button md-icon">
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<a href="intro_to_dynamic_programming/" class="md-footer__link md-footer__link--next" aria-label="Next: 14.1 Introduction to dynamic programming">
<a href="intro_to_dynamic_programming/" class="md-footer__link md-footer__link--next" aria-label="Next: 14.1 Introduction to Dynamic Programming">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
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<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
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<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 *
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<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>
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<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
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<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
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<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>
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<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
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<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
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<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
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<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
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<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
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<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
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<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
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<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
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<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
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<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
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<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
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<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
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<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
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<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
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<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
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<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
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<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
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<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
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<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
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<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
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<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
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<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
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<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
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<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
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<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
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<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
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<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
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<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
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<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
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<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
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Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
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12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
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12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
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12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
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12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
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13.1 Backtracking algorithms
13.1 Backtracking Algorithm
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13.2 Permutation problem
13.2 Permutations Problem
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13.3 Subset sum problem
13.3 Subset-Sum Problem
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13.4 n queens problem
13.4 N-Queens Problem
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Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
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Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
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14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
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14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
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14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
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14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
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14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
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14.6 Edit distance problem
14.6 Edit Distance Problem
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14.7 Summary
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15.1 Greedy algorithms
15.1 Greedy Algorithm
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15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
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15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
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15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
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16.1 Installation
16.1 Programming Environment Installation
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16.2 Contributing
16.2 Contributing Together
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16.3 Terminology
16.3 Terminology Table
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<h1 id="147-summary">14.7 &nbsp; Summary<a class="headerlink" href="#147-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>
<ul>
<li>Dynamic programming decomposes problems and improves computational efficiency by avoiding redundant computations through storing solutions of subproblems.</li>
<li>Without considering time, all dynamic programming problems can be solved using backtracking (brute force search), but the recursion tree has many overlapping subproblems, resulting in very low efficiency. By introducing a memorization list, it's possible to store solutions of all computed subproblems, ensuring that overlapping subproblems are only computed once.</li>
<li>Memorization search is a top-down recursive solution, whereas dynamic programming corresponds to a bottom-up iterative approach, akin to "filling out a table." Since the current state only depends on certain local states, we can eliminate one dimension of the dp table to reduce space complexity.</li>
<li>Decomposition of subproblems is a universal algorithmic approach, differing in characteristics among divide and conquer, dynamic programming, and backtracking.</li>
<li>Dynamic programming problems have three main characteristics: overlapping subproblems, optimal substructure, and no aftereffects.</li>
<li>If the optimal solution of the original problem can be constructed from the optimal solutions of its subproblems, it has an optimal substructure.</li>
<li>No aftereffects mean that the future development of a state depends only on the current state and not on all past states experienced. Many combinatorial optimization problems do not have this property and cannot be quickly solved using dynamic programming.</li>
<li>Dynamic programming decomposes problems and avoids redundant computation by storing the solutions to subproblems, thereby significantly improving computational efficiency.</li>
<li>Without considering time constraints, all dynamic programming problems can be solved using backtracking (brute force search), but the recursion tree contains a large number of overlapping subproblems, resulting in extremely low efficiency. By introducing a memo list, we can store the solutions to all computed subproblems, ensuring that overlapping subproblems are only computed once.</li>
<li>Memoization is a top-down recursive solution, while the corresponding dynamic programming is a bottom-up iterative solution, similar to "filling in a table". Since the current state only depends on certain local states, we can eliminate one dimension of the <span class="arithmatex">\(dp\)</span> table to reduce space complexity.</li>
<li>Subproblem decomposition is a general algorithmic approach, with different properties in divide and conquer, dynamic programming, and backtracking.</li>
<li>Dynamic programming problems have three major characteristics: overlapping subproblems, optimal substructure, and no aftereffects.</li>
<li>If the optimal solution to the original problem can be constructed from the optimal solutions to the subproblems, then it has optimal substructure.</li>
<li>No aftereffects means that for a given state, its future development is only related to that state and has nothing to do with all past states. Many combinatorial optimization problems do not have no aftereffects and cannot be quickly solved using dynamic programming.</li>
</ul>
<p><strong>Knapsack problem</strong></p>
<ul>
<li>The knapsack problem is one of the most typical dynamic programming problems, with variants including the 0-1 knapsack, unbounded knapsack, and multiple knapsacks.</li>
<li>The state definition of the 0-1 knapsack is the maximum value in a knapsack of capacity <span class="arithmatex">\(c\)</span> with the first <span class="arithmatex">\(i\)</span> items. Based on decisions not to include or to include an item in the knapsack, optimal substructures can be identified and state transition equations constructed. In space optimization, since each state depends on the state directly above and to the upper left, the list should be traversed in reverse order to avoid overwriting the upper left state.</li>
<li>In the unbounded knapsack problem, there is no limit on the number of each kind of item that can be chosen, thus the state transition for including items differs from the 0-1 knapsack. Since the state depends on the state directly above and to the left, space optimization should involve forward traversal.</li>
<li>The coin change problem is a variant of the unbounded knapsack problem, shifting from seeking the “maximum” value to seeking the “minimum” number of coins, thus the state transition equation should change <span class="arithmatex">\(\max()\)</span> to <span class="arithmatex">\(\min()\)</span>. From pursuing “not exceeding” the capacity of the knapsack to seeking exactly the target amount, thus use <span class="arithmatex">\(amt + 1\)</span> to represent the invalid solution of “unable to make up the target amount.”</li>
<li>Coin Change Problem II shifts from seeking the “minimum number of coins” to seeking the “number of coin combinations,” changing the state transition equation accordingly from <span class="arithmatex">\(\min()\)</span> to summation operator.</li>
<li>The knapsack problem is one of the most typical dynamic programming problems, with variants such as the 0-1 knapsack, unbounded knapsack, and multiple knapsack.</li>
<li>The state definition for the 0-1 knapsack is the maximum value among the first <span class="arithmatex">\(i\)</span> items in a knapsack of capacity <span class="arithmatex">\(c\)</span>. Based on the two decisions of not putting an item in the knapsack and putting it in, the optimal substructure can be identified and the state transition equation constructed. In space optimization, since each state depends on the state directly above and to the upper-left, the list needs to be traversed in reverse order to avoid overwriting the upper-left state.</li>
<li>The unbounded knapsack problem has no limit on the selection quantity of each type of item, so the state transition for choosing to put in an item differs from the 0-1 knapsack problem. Since the state depends on the state directly above and directly to the left, space optimization should use forward traversal.</li>
<li>The coin change problem is a variant of the unbounded knapsack problem. It changes from seeking the "maximum" value to seeking the "minimum" number of coins, so <span class="arithmatex">\(\max()\)</span> in the state transition equation should be changed to <span class="arithmatex">\(\min()\)</span>. It changes from seeking "not exceeding" the knapsack capacity to seeking "exactly" making up the target amount, so <span class="arithmatex">\(amt + 1\)</span> is used to represent the invalid solution of "unable to make up the target amount".</li>
<li>Coin change problem II changes from seeking the "minimum number of coins" to seeking the "number of coin combinations", so the state transition equation correspondingly changes from <span class="arithmatex">\(\min()\)</span> to a summation operator.</li>
</ul>
<p><strong>Edit distance problem</strong></p>
<ul>
<li>Edit distance (Levenshtein distance) measures the similarity between two strings, defined as the minimum number of editing steps needed to change one string into another, with editing operations including adding, deleting, or replacing.</li>
<li>The state definition for the edit distance problem is the minimum number of editing steps needed to change the first <span class="arithmatex">\(i\)</span> characters of <span class="arithmatex">\(s\)</span> into the first <span class="arithmatex">\(j\)</span> characters of <span class="arithmatex">\(t\)</span>. When <span class="arithmatex">\(s[i] \ne t[j]\)</span>, there are three decisions: add, delete, replace, each with their corresponding residual subproblems. From this, optimal substructures can be identified, and state transition equations built. When <span class="arithmatex">\(s[i] = t[j]\)</span>, no editing of the current character is necessary.</li>
<li>In edit distance, the state depends on the state directly above, to the left, and to the upper left. Therefore, after space optimization, neither forward nor reverse traversal can correctly perform state transitions. To address this, we use a variable to temporarily store the upper left state, making it equivalent to the situation in the unbounded knapsack problem, allowing for forward traversal after space optimization.</li>
<li>Edit distance (Levenshtein distance) is used to measure the similarity between two strings, defined as the minimum number of edit steps from one string to another, with edit operations including insert, delete, and replace.</li>
<li>The state definition for the edit distance problem is the minimum number of edit steps required to change the first <span class="arithmatex">\(i\)</span> characters of <span class="arithmatex">\(s\)</span> into the first <span class="arithmatex">\(j\)</span> characters of <span class="arithmatex">\(t\)</span>. When <span class="arithmatex">\(s[i] \ne t[j]\)</span>, there are three decisions: insert, delete, replace, each with corresponding remaining subproblems. From this, the optimal substructure can be identified and the state transition equation constructed. When <span class="arithmatex">\(s[i] = t[j]\)</span>, no edit is required for the current character.</li>
<li>In edit distance, the state depends on the state directly above, directly to the left, and to the upper-left, so after space optimization, neither forward nor reverse traversal can correctly perform state transitions. For this reason, we use a variable to temporarily store the upper-left state, thus transforming to a situation equivalent to the unbounded knapsack problem, allowing for forward traversal after space optimization.</li>
</ul>
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