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<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
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<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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<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
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<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
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<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 *
@@ -1430,7 +1430,7 @@
<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>
@@ -1530,7 +1530,7 @@
<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>
@@ -1665,7 +1665,7 @@
<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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<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 *
@@ -2137,7 +2137,7 @@
<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
@@ -3229,7 +3229,7 @@
<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
@@ -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
@@ -4306,14 +4306,14 @@
<p><a class="glightbox" href="../assets/covers/chapter_heap.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Heap" class="cover-image" src="../assets/covers/chapter_heap.jpg" /></a></p>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Heaps resemble mountains and their jagged peaks, layered and undulating, each with its unique form.</p>
<p>Each mountain peak rises and falls in scattered heights, yet the tallest always captures attention first.</p>
<p>Heaps are like mountain peaks, layered and undulating, each with its unique form.</p>
<p>The peaks rise and fall at varying heights, yet the tallest peak always catches the eye first.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="heap/">8.1 &nbsp; Heap</a></li>
<li><a href="build_heap/">8.2 &nbsp; Building a heap</a></li>
<li><a href="top_k/">8.3 &nbsp; Top-k problem</a></li>
<li><a href="build_heap/">8.2 &nbsp; Building a Heap</a></li>
<li><a href="top_k/">8.3 &nbsp; Top-K Problem</a></li>
<li><a href="summary/">8.4 &nbsp; Summary</a></li>
</ul>
en/chapter_heap/summary/index.html
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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
@@ -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 *
@@ -2137,7 +2137,7 @@
<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
@@ -2165,7 +2165,7 @@
<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
@@ -2249,7 +2249,7 @@
<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key review
1. &nbsp; Key Review
</span>
</a>
@@ -2397,7 +2397,7 @@
<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
@@ -2425,7 +2425,7 @@
<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
@@ -2564,7 +2564,7 @@
<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
@@ -2592,7 +2592,7 @@
<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
@@ -2620,7 +2620,7 @@
<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
@@ -2648,7 +2648,7 @@
<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
@@ -2676,7 +2676,7 @@
<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
@@ -2825,7 +2825,7 @@
<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
@@ -2853,7 +2853,7 @@
<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
@@ -2881,7 +2881,7 @@
<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
@@ -2909,7 +2909,7 @@
<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
@@ -2937,7 +2937,7 @@
<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
@@ -2965,7 +2965,7 @@
<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
@@ -2993,7 +2993,7 @@
<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
@@ -3021,7 +3021,7 @@
<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
@@ -3049,7 +3049,7 @@
<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
@@ -3077,7 +3077,7 @@
<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
@@ -3170,7 +3170,7 @@
<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
@@ -3192,7 +3192,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
</label>
@@ -3214,7 +3214,7 @@
<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
@@ -3242,7 +3242,7 @@
<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
@@ -3270,7 +3270,7 @@
<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
@@ -3298,7 +3298,7 @@
<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
@@ -3435,7 +3435,7 @@
<span class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
@@ -3463,7 +3463,7 @@
<span class="md-ellipsis">
13.2 Permutation problem
13.2 Permutations Problem
@@ -3491,7 +3491,7 @@
<span class="md-ellipsis">
13.3 Subset sum problem
13.3 Subset-Sum Problem
@@ -3519,7 +3519,7 @@
<span class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
@@ -3616,7 +3616,7 @@
<span class="md-ellipsis">
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
@@ -3638,7 +3638,7 @@
<span class="md-nav__icon md-icon"></span>
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
</label>
@@ -3660,7 +3660,7 @@
<span class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
@@ -3688,7 +3688,7 @@
<span class="md-ellipsis">
14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
@@ -3716,7 +3716,7 @@
<span class="md-ellipsis">
14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
@@ -3744,7 +3744,7 @@
<span class="md-ellipsis">
14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
@@ -3772,7 +3772,7 @@
<span class="md-ellipsis">
14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
@@ -3800,7 +3800,7 @@
<span class="md-ellipsis">
14.6 Edit distance problem
14.6 Edit Distance Problem
@@ -3937,7 +3937,7 @@
<span class="md-ellipsis">
15.1 Greedy algorithms
15.1 Greedy Algorithm
@@ -3965,7 +3965,7 @@
<span class="md-ellipsis">
15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
@@ -3993,7 +3993,7 @@
<span class="md-ellipsis">
15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
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<span class="md-ellipsis">
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="84-summary">8.4 &nbsp; Summary<a class="headerlink" href="#84-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>A heap is a complete binary tree that can be categorized as either a max heap or a min heap based on its building property, where the top element of a max heap is the largest and the top element of a min heap is the smallest.</li>
<li>A priority queue is defined as a queue with dequeue priority, usually implemented using a heap.</li>
<li>Common operations of a heap and their corresponding time complexities include: element insertion into the heap <span class="arithmatex">\(O(\log n)\)</span>, removing the top element from the heap <span class="arithmatex">\(O(\log n)\)</span>, and accessing the top element of the heap <span class="arithmatex">\(O(1)\)</span>.</li>
<li>A complete binary tree is well-suited to be represented by an array, thus heaps are commonly stored using arrays.</li>
<li>Heapify operations are used to maintain the properties of the heap and are used in both heap insertion and removal operations.</li>
<li>The time complexity of building a heap given an input of <span class="arithmatex">\(n\)</span> elements can be optimized to <span class="arithmatex">\(O(n)\)</span>, which is highly efficient.</li>
<li>A heap is a complete binary tree that can be categorized as a max heap or min heap based on its property. The heap top element of a max heap (min heap) is the largest (smallest).</li>
<li>A priority queue is defined as a queue with priority sorting, typically implemented using heaps.</li>
<li>Common heap operations and their corresponding time complexities include: element insertion <span class="arithmatex">\(O(\log n)\)</span>, heap top element removal <span class="arithmatex">\(O(\log n)\)</span>, and accessing the heap top element <span class="arithmatex">\(O(1)\)</span>.</li>
<li>Complete binary trees are well-suited for array representation, so we typically use arrays to store heaps.</li>
<li>Heapify operations are used to maintain the heap property and are employed in both element insertion and removal operations.</li>
<li>The time complexity of building a heap with <span class="arithmatex">\(n\)</span> input elements can be optimized to <span class="arithmatex">\(O(n)\)</span>, which is highly efficient.</li>
<li>Top-k is a classic algorithm problem that can be efficiently solved using the heap data structure, with a time complexity of <span class="arithmatex">\(O(n \log k)\)</span>.</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>: Is the "heap" in data structures the same concept as the "heap" in memory management?</p>
<p>The two are not the same concept, even though they are both referred to as "heap". The heap in computer system memory is part of dynamic memory allocation, where the program can use it to store data during execution. The program can request a certain amount of heap memory to store complex structures like objects and arrays. When the allocated data is no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, the management and usage of heap memory demands more caution, as improper use may lead to memory leaks and dangling pointers.</p>
<p><strong>Q</strong>: Are the "heap" in data structures and the "heap" in memory management the same concept?</p>
<p>The two are not the same concept; they just happen to share the name "heap." The heap in computer system memory is part of dynamic memory allocation, where programs can use it to store data during runtime. Programs can request a certain amount of heap memory to store complex structures such as objects and arrays. When this data is no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, heap memory management and usage require more caution, as improper use can lead to issues such as memory leaks and dangling pointers.</p>
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