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<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>
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<style>:root{--md-text-font:"Lato";--md-code-font:"JetBrains Mono"}</style>
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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
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<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 *
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<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>
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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
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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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<label class="md-nav__link md-nav__link--active" for="__toc">
<span class="md-ellipsis">
15.5 Summary
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
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</a>
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
<label class="md-nav__title" for="__toc">
<span class="md-nav__icon md-icon"></span>
Table of contents
</label>
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key Review
</span>
</a>
</li>
</ul>
</nav>
</li>
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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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<label class="md-nav__title" for="__toc">
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Table of contents
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key Review
</span>
</a>
</li>
</ul>
</nav>
</div>
</div>
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<!-- Page content -->
<h1 id="155-summary">15.5 &nbsp; Summary<a class="headerlink" href="#155-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>Greedy algorithms are often used to solve optimization problems, where the principle is to make locally optimal decisions at each decision stage in order to achieve a globally optimal solution.</li>
<li>Greedy algorithms iteratively make one greedy choice after another, transforming the problem into a smaller sub-problem with each round, until the problem is resolved.</li>
<li>Greedy algorithms are not only simple to implement but also have high problem-solving efficiency. Compared to dynamic programming, greedy algorithms generally have a lower time complexity.</li>
<li>In the problem of coin change, greedy algorithms can guarantee the optimal solution for certain combinations of coins; for others, however, the greedy algorithm might find a very poor solution.</li>
<li>Problems suitable for greedy algorithm solutions possess two main properties: greedy-choice property and optimal substructure. The greedy-choice property represents the effectiveness of the greedy strategy.</li>
<li>For some complex problems, proving the greedy-choice property is not straightforward. Contrarily, proving the invalidity is often easier, such as with the coin change problem.</li>
<li>Solving greedy problems mainly consists of three steps: problem analysis, determining the greedy strategy, and proving correctness. Among these, determining the greedy strategy is the key step, while proving correctness often poses the challenge.</li>
<li>The fractional knapsack problem builds on the 0-1 knapsack problem by allowing the selection of a part of the items, hence it can be solved using a greedy algorithm. The correctness of the greedy strategy can be proved by contradiction.</li>
<li>The maximum capacity problem can be solved using the exhaustive method, with a time complexity of <span class="arithmatex">\(O(n^2)\)</span>. By designing a greedy strategy, each round moves inwardly shortening the board, optimizing the time complexity to <span class="arithmatex">\(O(n)\)</span>.</li>
<li>In the problem of maximum product after cutting, we deduce two greedy strategies: integers <span class="arithmatex">\(\geq 4\)</span> should continue to be cut, with the optimal cutting factor being <span class="arithmatex">\(3\)</span>. The code includes power operations, and the time complexity depends on the method of implementing power operations, generally being <span class="arithmatex">\(O(1)\)</span> or <span class="arithmatex">\(O(\log n)\)</span>.</li>
<li>Greedy algorithms are typically used to solve optimization problems. The principle is to make locally optimal decisions at each decision stage in hopes of obtaining a globally optimal solution.</li>
<li>Greedy algorithms iteratively make one greedy choice after another, transforming the problem into a smaller subproblem in each round, until the problem is solved.</li>
<li>Greedy algorithms are not only simple to implement, but also have high problem-solving efficiency. Compared to dynamic programming, greedy algorithms typically have lower time complexity.</li>
<li>In the coin change problem, for certain coin combinations, greedy algorithms can guarantee finding the optimal solution; for other coin combinations, however, greedy algorithms may find very poor solutions.</li>
<li>Problems suitable for solving with greedy algorithms have two major properties: greedy choice property and optimal substructure. The greedy choice property represents the effectiveness of the greedy strategy.</li>
<li>For some complex problems, proving the greedy choice property is not simple. Relatively speaking, disproving it is easier, such as in the coin change problem.</li>
<li>Solving greedy problems mainly consists of three steps: problem analysis, determining the greedy strategy, and correctness proof. Among these, determining the greedy strategy is the core step, and correctness proof is often the difficult point.</li>
<li>The fractional knapsack problem, based on the 0-1 knapsack problem, allows selecting a portion of items, and therefore can be solved using greedy algorithms. The correctness of the greedy strategy can be proven using proof by contradiction.</li>
<li>The max capacity problem can be solved using exhaustive enumeration with time complexity <span class="arithmatex">\(O(n^2)\)</span>. By designing a greedy strategy to move the short partition inward in each round, the time complexity can be optimized to <span class="arithmatex">\(O(n)\)</span>.</li>
<li>In the max product cutting problem, we successively derive two greedy strategies: integers <span class="arithmatex">\(\geq 4\)</span> should all continue to be split, and the optimal splitting factor is <span class="arithmatex">\(3\)</span>. The code includes exponentiation operations, and the time complexity depends on the implementation method of exponentiation, typically being <span class="arithmatex">\(O(1)\)</span> or <span class="arithmatex">\(O(\log n)\)</span>.</li>
</ul>
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15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
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15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
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