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<title>Chapter 6.   Hash table - Hello Algo</title>
<title>Chapter 6.   Hashing - Hello Algo</title>
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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>
<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>
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<div data-md-component="skip">
<a href="#chapter-6-hash-table" class="md-skip">
<a href="#chapter-6-hashing" 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">
Chapter 6. &nbsp; Hash table
Chapter 6. &nbsp; Hashing
</span>
</div>
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<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
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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
@@ -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
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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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<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
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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
@@ -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
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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
@@ -4302,18 +4302,18 @@
<!-- Page content -->
<h1 id="chapter-6-hash-table">Chapter 6. &nbsp; Hash table<a class="headerlink" href="#chapter-6-hash-table" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_hashing.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Hash table" class="cover-image" src="../assets/covers/chapter_hashing.jpg" /></a></p>
<h1 id="chapter-6-hashing">Chapter 6. &nbsp; Hashing<a class="headerlink" href="#chapter-6-hashing" title="Permanent link">&para;</a></h1>
<p><a class="glightbox" href="../assets/covers/chapter_hashing.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Hashing" class="cover-image" src="../assets/covers/chapter_hashing.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>In the world of computing, a hash table is akin to an intelligent librarian.</p>
<p>It understands how to compute index numbers, enabling swift retrieval of the desired book.</p>
<p>In the world of computing, a hash table is like a clever librarian.</p>
<p>They know how to calculate call numbers, enabling them to quickly locate the target book.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="hash_map/">6.1 &nbsp; Hash table</a></li>
<li><a href="hash_collision/">6.2 &nbsp; Hash collision</a></li>
<li><a href="hash_algorithm/">6.3 &nbsp; Hash algorithm</a></li>
<li><a href="hash_map/">6.1 &nbsp; Hash Table</a></li>
<li><a href="hash_collision/">6.2 &nbsp; Hash Collision</a></li>
<li><a href="hash_algorithm/">6.3 &nbsp; Hash Algorithm</a></li>
<li><a href="summary/">6.4 &nbsp; Summary</a></li>
</ul>
@@ -4362,7 +4362,7 @@ aria-label="Footer"
<a
href="hash_map/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 6.1 Hash table"
aria-label="Next: 6.1 Hash Table"
rel="next"
>
<div class="md-footer__title">
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Next
</span>
<div class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
</div>
</div>
<div class="md-footer__button md-icon">
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<a href="hash_map/" class="md-footer__link md-footer__link--next" aria-label="Next: 6.1 Hash table">
<a href="hash_map/" class="md-footer__link md-footer__link--next" aria-label="Next: 6.1 Hash Table">
<div class="md-footer__title">
<span class="md-footer__direction">
Next
</span>
<div class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
</div>
</div>
<div class="md-footer__button md-icon">
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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>
<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
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<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
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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>
@@ -1530,7 +1530,7 @@
<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
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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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<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key review
1. &nbsp; Key Review
</span>
</a>
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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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<span class="md-nav__icon md-icon"></span>
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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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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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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1. &nbsp; Key review
1. &nbsp; Key Review
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<h1 id="64-summary">6.4 &nbsp; Summary<a class="headerlink" href="#64-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>Given an input <code>key</code>, a hash table can retrieve the corresponding <code>value</code> in <span class="arithmatex">\(O(1)\)</span> time, which is highly efficient.</li>
<li>Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table.</li>
<li>The hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket and retrieval of the <code>value</code>.</li>
<li>Two different keys may end up with the same array index after hashing, leading to erroneous query results. This phenomenon is known as hash collision.</li>
<li>The larger the capacity of the hash table, the lower the probability of hash collisions. Therefore, hash table resizing can mitigate hash collisions. Similar to array resizing, hash table resizing is costly.</li>
<li>The load factor, defined as the number of elements divided by the number of buckets, reflects the severity of hash collisions and is often used as a condition to trigger hash table resizing.</li>
<li>Chaining addresses hash collisions by converting each element into a linked list, storing all colliding elements in the same list. However, excessively long lists can reduce query efficiency, which can be improved by converting the lists into red-black trees.</li>
<li>Open addressing handles hash collisions through multiple probes. Linear probing uses a fixed step size but it cannot delete elements and is prone to clustering. Multiple hashing uses several hash functions for probing which reduces clustering compared to linear probing but increases computational overhead.</li>
<li>Different programming languages adopt various hash table implementations. For example, Java's <code>HashMap</code> uses chaining, while Python's <code>dict</code> employs open addressing.</li>
<li>The larger the capacity of the hash table, the lower the probability of hash collisions. Therefore, hash table expansion can mitigate hash collisions. Similar to array expansion, hash table expansion is costly.</li>
<li>The load factor, defined as the number of elements divided by the number of buckets, reflects the severity of hash collisions and is often used as a condition to trigger hash table expansion.</li>
<li>Separate chaining addresses hash collisions by converting each element into a linked list, storing all colliding elements in the same linked list. However, excessively long linked lists can reduce query efficiency, which can be improved by converting the linked lists into red-black trees.</li>
<li>Open addressing handles hash collisions through multiple probing. Linear probing uses a fixed step size but cannot delete elements and is prone to clustering. Double hashing uses multiple hash functions for probing, which reduces clustering compared to linear probing but increases computational overhead.</li>
<li>Different programming languages adopt various hash table implementations. For example, Java's <code>HashMap</code> uses separate chaining, while Python's <code>dict</code> employs open addressing.</li>
<li>In hash tables, we desire hash algorithms with determinism, high efficiency, and uniform distribution. In cryptography, hash algorithms should also possess collision resistance and the avalanche effect.</li>
<li>Hash algorithms typically use large prime numbers as moduli to ensure uniform distribution of hash values and reduce hash collisions.</li>
<li>Hash algorithms typically use large prime numbers as moduli to maximize the uniform distribution of hash values and reduce hash collisions.</li>
<li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file integrity checks, while SHA-2 is commonly used in secure applications and protocols.</li>
<li>Programming languages usually provide built-in hash algorithms for data types to calculate bucket indices in hash tables. Generally, only immutable objects are hashable.</li>
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<p><strong>Q</strong>: Why can hash tables be more efficient than arrays, linked lists, or binary trees, even though hash tables are implemented using these structures?</p>
<p>Firstly, hash tables have higher time efficiency but lower space efficiency. A significant portion of memory in hash tables remains unused.</p>
<p>Secondly, hash tables are only more time-efficient in specific use cases. If a feature can be implemented with the same time complexity using an array or a linked list, it's usually faster than using a hash table. This is because the computation of the hash function incurs overhead, making the constant factor in the time complexity larger.</p>
<p>Lastly, the time complexity of hash tables can degrade. For example, in chaining, we perform search operations in a linked list or red-black tree, which still risks degrading to <span class="arithmatex">\(O(n)\)</span> time.</p>
<p><strong>Q</strong>: Does multiple hashing also have the flaw of not being able to delete elements directly? Can space marked as deleted be reused?</p>
<p>Multiple hashing is a form of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they require marking elements as deleted. Marked spaces can be reused. When inserting new elements into the hash table, and the hash function points to a position marked as deleted, that position can be used by the new element. This maintains the probing sequence of the hash table while ensuring efficient use of space.</p>
<p>Lastly, the time complexity of hash tables can degrade. For example, in separate chaining, we perform search operations in a linked list or red-black tree, which still risks degrading to <span class="arithmatex">\(O(n)\)</span> time.</p>
<p><strong>Q</strong>: Does double hashing also have the flaw of not being able to delete elements directly? Can space marked as deleted be reused?</p>
<p>Double hashing is a form of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they require marking elements as deleted. Marked spaces can be reused. When inserting new elements into the hash table, and the hash function points to a position marked as deleted, that position can be used by the new element. This maintains the probing sequence of the hash table while ensuring efficient use of space.</p>
<p><strong>Q</strong>: Why do hash collisions occur during the search process in linear probing?</p>
<p>During the search process, the hash function points to the corresponding bucket and key-value pair. If the <code>key</code> doesn't match, it indicates a hash collision. Therefore, linear probing will search downwards at a predetermined step size until the correct key-value pair is found or the search fails.</p>
<p><strong>Q</strong>: Why can resizing a hash table alleviate hash collisions?</p>
<p>The last step of a hash function often involves taking the modulo of the array length <span class="arithmatex">\(n\)</span>, to keep the output within the array index range. When resizing, the array length <span class="arithmatex">\(n\)</span> changes, and the indices corresponding to the keys may also change. Keys that were previously mapped to the same bucket might be distributed across multiple buckets after resizing, thereby mitigating hash collisions.</p>
<p>During the search process, the hash function points to the corresponding bucket and key-value pair. If the <code>key</code> doesn't match, it indicates a hash collision. Therefore, linear probing will search downward at a predetermined step size until the correct key-value pair is found or the search fails.</p>
<p><strong>Q</strong>: Why can expanding a hash table alleviate hash collisions?</p>
<p>The last step of a hash function often involves taking the modulo of the array length <span class="arithmatex">\(n\)</span>, to keep the output within the array index range. When expanding, the array length <span class="arithmatex">\(n\)</span> changes, and the indices corresponding to the keys may also change. Keys that were previously mapped to the same bucket might be distributed across multiple buckets after expansion, thereby mitigating hash collisions.</p>
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