Heap Sort in Data Structures: Key Concepts with Examples
By Rohit Sharma
Updated on Jun 17, 2025 | 16 min read | 12.53K+ views
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By Rohit Sharma
Updated on Jun 17, 2025 | 16 min read | 12.53K+ views
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Did you know? Heap Sort guarantees O(n log n) worst-case time and uses only O(1) space, making it ideal for memory-constrained and real-time systems where consistent performance matters. |
Heap Sort in data structures is a comparison-based sorting algorithm that organizes elements using a binary heap structure. With a worst-case time complexity of O(n log n) and constant space usage, it's well-suited for systems where memory efficiency and predictable performance are essential—such as priority queues, embedded systems, and scheduling algorithms.
In this blog, we’ll explain how Heap Sort works, outline its advantages and limitations, and explore practical use cases where it's commonly applied.
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Heap sort in data structures is a comparison-based sorting algorithm using a binary heap data structure to sort elements. It is an improvised version of selection sort, as it repeatedly selects the largest (or smallest) element using a heap instead of a linear scan. Heapsort divides the given input into sorted and unsorted regions, like selection sort. It keeps shrinking the unsorted region by removing the largest or smallest element, depending on the sorting order, and keeps adding them to the sorted region.
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Before proceeding further, let's understand a Complete Binary Tree and Heapify.
Before diving into Heap Sort, it’s important to understand two key concepts: heapify and binary trees, especially complete binary trees. These form the backbone of how Heap Sort in data structures works. Let’s break them down.
Heapify is the process of transforming a part of a binary tree into a heap, either a Max-Heap or a Min-Heap, depending on the context. In Heap Sort, the heap structure may break after placing the largest (or smallest) element at the top. To fix this, we use heapify to restore the heap property.
Example: In a Max-Heap, each parent node should be greater than or equal to its children. If this isn’t true, heapify compares the parent with its children, swaps them if needed, and keeps doing this down the subtree until the property is restored.
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A binary tree is a tree data structure in which each node has at most two children, usually the left and right. Heaps are a binary tree, so understanding this structure is key to grasping how Heap Sort in data structures works.
There are several types of binary trees, such as full binary trees, perfect binary trees, complete binary trees, and balanced binary trees. However, the complete binary tree is the most relevant for understanding Heap Sort.
A Binary Tree is Complete if all the levels are completely filled except possibly the last level, and the last level has all nodes as far left as possible. A complete binary tree is just like a full binary tree, but with two major differences:
Example 1:
Example2:
Example3:
Also Read: 5 Types of Binary Trees: Key Concepts, Structures, and Real-World Applications in 2025
Now that you understand a heap, heapify, and binary tree, let’s explore how Heap Sort works in data structures.
Heap Sort in data structures uses the heap data structure, a special type of complete binary tree. In a heap, each parent node is either greater than or equal to (in a Max-Heap) or less than or equal to (in a Min-Heap) its child nodes.
To be considered a heap, a binary tree must satisfy two main conditions:
In Heap Sort, a Max-Heap is commonly used to sort elements in ascending order. The algorithm repeatedly moves the largest element (the root) to the end of the array, then restores the heap structure using heapify on the remaining elements.
Heap Sort in data structures uses a set of simple formulas to treat the array as a complete binary tree, which is central to how the sorting process works. Instead of building an actual tree, the algorithm works directly with array indices to simulate the structure of a binary heap. This is possible because a complete binary tree has a simple and efficient structure that allows us to represent parent-child relationships using array indices.
Here’s how it works:
This mapping makes it easy to traverse the tree, apply heapify, and perform swaps without needing explicit tree nodes or pointers. It’s one of the reasons Heap Sort is both efficient and space-optimized.
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Let’s look at how Heap Sort is implemented using a Max Heap to sort elements efficiently.
To sort elements in ascending order, Heap Sort in data structures relies on a Max-Heap. A key part of the process is a function called max_heapify(A, i), which takes an array A and an index i as input. It ensures that the subtree rooted at index i satisfies the Max-Heap property, meaning the parent node is greater than or equal to its children.
If the property is violated, max_heapify swaps the node with the largest of its children and recursively fixes the affected subtree. This step is repeated throughout the sorting process to maintain the heap structure after each extraction, ensuring the algorithm proceeds correctly.
const maxHeapify = (arr, n, i) => {
let largest = i;
let l = 2 * i + 1; //left child index
let r = 2 * i + 2; //right child index
//If left child is smaller than root
if (l < n && arr[l] > arr[largest]) {
largest = l;
}
// If right child is smaller than smallest so far
if (r < n && arr[r] > arr[largest]) {
largest = r;
}
// If smallest is not root
if (largest != i) {
let temp = arr[i];
arr[i] = arr[largest];
arr[largest] = temp;
// Recursively heapify the affected sub-tree
maxHeapify(arr, n, largest);
}
}
// main function to do heap sort
const heapSort = (arr, n) => {
// Build heap (rearrange array)
for (let i = parseInt(n / 2 - 1); i >= 0; i--) {
maxHeapify(arr, n, i);
}
// One by one extract an element from heap
for (let i = n - 1; i >= 0; i--) {
// Move current root to end
let temp = arr[0];
arr[0] = arr[i];
arr[i] = temp;
// call max heapify on the reduced heap
maxHeapify(arr, i, 0);
}
}
Code Explanation:
Input:
const arr = [4, 10, 3, 5, 1];
heapSort(arr, arr.length);
console.log(arr);
Output:
[1, 3, 4, 5, 10]
While Max Heap is commonly used, Heap Sort in data structures can also be implemented using a Min Heap. Let’s see how that works.
To sort elements in descending order, Heap Sort uses a Min Heap. The min_heapify(A, i) function ensures the subtree at index i maintains the Min Heap property where the parent is smaller than its children. If not, it swaps with the smallest child and recursively fixes the structure to keep the heap valid during sorting.
const minHeapify = (arr, n, i) => {
let smallest = i;
let l = 2 * i + 1; //left child index
let r = 2 * i + 2; //right child index
//If left child is smaller than root
if (l < n && arr[l] < arr[smallest]) {
smallest = l;
}
// If right child is smaller than smallest so far
if (r < n && arr[r] < arr[smallest]) {
smallest = r;
}
// If smallest is not root
if (smallest != i) {
let temp = arr[i];
arr[i] = arr[smallest];
arr[smallest] = temp;
// Recursively heapify the affected sub-tree
minHeapify(arr, n, smallest);
}
}
// main function to do heap sort
const heapSort = (arr, n) => {
// Build heap (rearrange array)
for (let i = parseInt(n / 2 - 1); i >= 0; i--) {
minHeapify(arr, n, i);
}
// One by one extract an element from heap
for (let i = n - 1; i >= 0; i--) {
// Move current root to end
let temp = arr[0];
arr[0] = arr[i];
arr[i] = temp;
// call min heapify on the reduced heap
minHeapify(arr, i, 0);
}
}
Code Explanation:
Let me know if you'd like this tailored for min-to-max order or max-to-min as well.
Input:
const arr = [4, 6, 3, 2, 9];
heapSort(arr, arr.length);
console.log(arr);
Output:
[9, 6, 4, 3, 2]
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Also Read: Time and Space Complexity of Binary Search Explained
After exploring how Heap Sort works with a Min Heap, let’s analyze its time and space complexity to understand its overall performance.
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Understanding the complexity of Heap Sort is crucial to evaluating when and where to use it. Heap Sort offers a strong balance between performance and memory efficiency. Unlike Quick Sort, which can degrade to O(n²) in the worst case, Heap Sort in data structures consistently maintains an O(n log n) time complexity, making it a reliable choice for large datasets.
Additionally, its in-place sorting nature with O(1) space complexity makes it ideal for systems with limited memory. Here is a breakdown of time and space complexities, and an explanation of why Heap Sort behaves the way it does.
Time Complexity
Case |
Time Complexity |
Best | O(n log n) |
Worst | O(n log n) |
Average | O(n log n) |
Also Read: Algorithm Complexity and Data Structure: Types of Time Complexity
Why O(n log n)?
Heap Sort in data structures has a consistent time complexity of O(n log n) across all cases — best, average, and worst. This happens due to two main steps in the algorithm:
Since both steps combined are O(n) + O(n log n), the dominant term is O(n log n).
Space Complexity
Heap Sort is an in-place sorting algorithm, meaning it does not require additional space to sort the elements. The only extra memory used is for:
These do not scale with input size n, so the space complexity is O(1).
This makes Heap Sort an excellent choice for memory-constrained environments where predictable performance and low space usage are critical.
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Also Read: Quick Sort Algorithm: Time Complexity and Examples
Now that you understand how Heap Sort works and performs, let’s explore where this algorithm is used in real-world development.
Heap Sort in data structures is more than just a sorting algorithm. Its underlying heap data structure makes it worthwhile in various computational problems that require efficient retrieval of maximum or minimum elements. Below are several key areas where Heap Sort proves valuable.
Understanding where Heap Sort in data structures shines can help you decide when to use it. Let’s look at its key advantages.
Heap Sort is a comparison-based, in-place sorting algorithm that offers consistent time complexity and memory efficiency. It's handy in systems where predictable performance and low space usage are essential. However, like any algorithm, it comes with trade-offs that developers should know before choosing it for real-world applications.
Here are the key advantages and disadvantages of using Heap Sort:
Advantages of Heap Sort
Disadvantages of Heap Sort
Heap Sort is a comparison-based algorithm with consistent O(n log n) time complexity and in-place sorting, making it efficient for large datasets. It uses a complete binary heap structure to organize elements, ensuring memory efficiency. Though not stable or cache-friendly, it delivers predictable performance. This makes it ideal for use cases like priority queues and top-k element selection.
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References:
https://blog.heycoach.in/heap-sort-vs-merge-sort/
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