Heap Sort in Data Structures: Key Concepts with Examples

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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What is Heap Sort in Data Structures? Core concepts

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.

• It is better than selection sort because, rather than wasting time on a linear-time scan of the unsorted region, it maintains a heap data structure to find the largest or smallest element quickly.
• Heap Sort in data structures is an in-place sorting algorithm. It doesn't require extra space. However, it is not stable. The relative order of equal elements can change during heapify.
• It is a tree-based sort that takes an array as an input and converts it to a tree using a certain formula, which we discuss later.
• When implementing Heap Sort, you can use either a Max-Heap or a Min-Heap, depending on the desired sorting order. However, Heap Sort default uses a Max-Heap to sort elements in ascending order.

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Before proceeding further, let's understand a Complete Binary Tree and Heapify.

Prerequisites: What Is Heapify and a Binary Tree?

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.

What Is Heapify?

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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What Is a Binary Tree?

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.

What is a Complete Binary Tree?

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:

• Every level except the last level must be completely filled.
• All the leaf elements must lean towards the left.
• The last leaf element might not have a right sibling, i.e., a complete binary tree doesn't have to be a full binary tree.

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.

How Does Heap Sort in Data Structures Work?

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:

1. It must be a complete binary tree. All levels are filled except possibly the last, which is filled from left to right.
2. Based on the sorting requirement, it must satisfy the heap property – either Max-Heap or Min-Heap.

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.

Array to Tree Conversion in Heap Sort

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:

• For any element at index i in the array:
  • The left child is at index 2i + 1
  • The right child is at index 2i + 2
• To find the parent of any element at index i:
  • Use the formula: floor((i - 1) / 2)

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.

Implementation of Heap Sort with Max Heap in Java

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:

• Max Heapify Function: The maxHeapify function ensures the subtree rooted at a given index i maintains the max-heap property. It assumes the current node is the largest and then compares it with its left and right children. If any child is larger, it updates the largest variable. If the largest is not the current node, it swaps values and recursively heapifies the affected subtree. This helps maintain a valid max heap throughout the array.
• Left and Right Child Indexing: In a binary heap represented as an array, the left child of a node at index i is located at 2 * i + 1, and the right child is at 2 * i + 2. These formulas navigate the heap structure without creating actual tree nodes.
• Building the Max Heap: The array must be transformed into a max heap before sorting. This is done by calling maxHeapify from the last non-leaf node (n / 2 - 1) up to the root. This ensures all heap levels satisfy the max-heap condition, which is crucial for the sorting phase.
• Sorting by Extraction: The sorting begins once the max heap is built. The largest element (at the root) is swapped with the last element in the array. The heap size is reduced by 1, and maxHeapify is called on the new root to restore the heap property. This process repeats, gradually moving the largest remaining element to the end of the unsorted portion.
• In-Place Sorting: Heap sort does not require additional memory for another array, making it an in-place sorting algorithm. All operations (heap building and extraction) are done within the input array.

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.

Implementation of Heap Sort with Min Heap in Java

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:

• Min Heapify Function: The minHeapify function maintains the min-heap property for a subtree rooted at a given index i. It starts by assuming the current node is the smallest. Then, it compares the node with its left and right children. If either child is smaller than the current smallest, it updates the smallest index. If the smallest is not the root, the values are swapped, and the function is recursively called on the affected subtree to ensure the entire subtree satisfies the min-heap condition.
• Left and Right Child Indexing: In a binary heap implemented using an array, the left child of a node at index i is located at 2 * i + 1, and the right child is at 2 * i + 2. These formulas allow you to navigate the heap using array indices without needing explicit tree structures.
• Building the Min Heap: The array is converted into a min heap before sorting begins. This is done by calling minHeapify on all non-leaf nodes starting from n / 2 - 1 down to the root. This ensures that every subtree satisfies the min-heap property, with the smallest element at the top.
• Sorting by Extraction: After the min heap is built, the smallest element (at index 0) is repeatedly swapped with the last element of the unsorted region. Then, the heap size is reduced by one, and minHeapify is called on the root to maintain the min-heap structure. This process continues until the entire array is sorted in descending order (since the smallest elements are pushed to the end).
• In-Place Sorting: This implementation of heap sort does not use any extra space, making it an in-place sorting algorithm. All sorting operations happen within the input array, with constant space complexity.

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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Complexity Analysis of Heap Sort Algorithm

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

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:

1. Building the Max-Heap: This step processes n elements and takes O(n) time.
2. Heapifying during sort: After building the heap, we repeatedly remove the root (maximum element) and call heapify to restore the heap.
   • Each heapify operation takes O(log n) time (because the height of the heap is log n)
   • This step runs n times (once for each element), so the total time is O(n log n)

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:

• A few loop counters
• Temporary variables like temp, index, or largest

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.

Practical Applications of the Heap Sort Algorithm

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.

• Sorting: Efficient for ascending and descending order with a guaranteed O(n log n) time complexity, making it suitable for large datasets.
• Priority Queues: Heaps are ideal for implementing priority queues, supporting fast insertion and extraction based on priority.
• Top-K Selection: Used to find the top k largest or smallest elements in a dataset with a time complexity of O(k log n).
• Median Maintenance: It helps stream data scenarios. Two heaps (min-heap and max-heap) can dynamically find the median. While Heap Sort isn't used directly in streaming, its underlying heap structure powers dynamic median maintenance with two heaps.
• External Sorting: Suitable for scenarios where data cannot fit entirely into memory, such as sorting very large files on disk.

Understanding where Heap Sort in data structures shines can help you decide when to use it. Let’s look at its key advantages.

Pros and Cons of the Heap Sort Algorithm

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

• Efficiency: Guarantees a consistent time complexity of O(n log n) in all cases—best, average, and worst.
• Space Efficient: Performs in-place sorting using constant auxiliary space, avoiding extra memory usage.
• Predictable Performance: Unlike Quick Sort, which can degrade to O(n²) in the worst case, Heap Sort remains stable at O(n log n).
• In-Place Sorting: Sorts the data within the original array, eliminating the need for additional storage.
• Conceptual Simplicity: While it involves heap operations, the core logic is more straightforward to grasp than more complex algorithms like Merge Sort or Quick Sort with pivoting.
• Supports Partial Sorting: It can be adapted to extract top-k elements from a dataset efficiently and is functional in leaderboard systems or priority queues.

Disadvantages of Heap Sort

• Not Stable: Fails to maintain the relative order of equal elements, a limitation when stability is essential.
• Poor Cache Performance: Due to non-sequential memory access patterns, it may underperform on large arrays compared to algorithms like Merge Sort.
• Tricky Implementation: Requires precise handling of heapify and index calculations, which can introduce bugs if not implemented carefully.
• Not Adaptive: Does not take advantage of partially sorted input—always performs the full sorting process.
• Perceived Space Overhead: Though in-place by design, certain implementations may temporarily use extra space during heap construction.

How Can upGrad Help You Excel in Data Science?

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/