**What Is Heap Sort Algorithm in Data Structure?**

Heap sort is a comparison-based sorting algorithm that falls under the category of selection sorts. It is an efficient sorting algorithm that works by dividing the input into a sorted and an unsorted region, and iteratively shrinking the unsorted region by extracting the largest element from it and inserting it into the sorted region.

## Understanding Heaps

Before diving into heap sort, it’s important to understand what heaps are. A heap is a complete binary tree that satisfies the heap property.

The heap property states that for every node in the tree, its value must be greater than or equal to (in a max heap) or less than or equal to (in a min heap) its children’s values.

One common representation of a heap is an array where each element corresponds to a node in the binary tree. The root of the tree is represented by the first element of the array, and for any given node at index i, its left child is at index 2i+1 and its right child is at index 2i+2.

## The Heap Sort Algorithm

The heap sort algorithm can be divided into two main steps: building a max heap and repeatedly extracting the maximum element from it.

### Building a Max Heap

To build a max heap, we start with an unsorted array and perform a series of swaps to satisfy the heap property. This process begins from the last non-leaf node and works its way up to the root of the tree.

Starting from index n/2-1 (where n is the size of the array), we compare each node with its children nodes. If any child has a larger value, we swap it with the parent node.

This ensures that the largest element in each subtree is at the root of that subtree.

After performing these swaps for all non-leaf nodes, we end up with a max heap where the largest element is at the root.

### Extracting the Maximum Element

Once we have a max heap, we repeatedly extract the maximum element and place it at the end of the unsorted region of the array. We then reheapify the remaining elements in the unsorted region to maintain the heap property.

To extract the maximum element, we swap it with the last element in the unsorted region and reduce the size of the unsorted region by one. We then perform a down-heap operation starting from the root to restore the max heap property.

This process continues until all elements are sorted.

## Complexity Analysis

Heap sort has a time complexity of O(n log n) for both average and worst-case scenarios. The space complexity is O(1) as it doesn’t require any additional data structures besides the input array itself.

## Conclusion

Heap sort is an efficient sorting algorithm that leverages binary heaps to divide an array into a sorted and an unsorted region. By repeatedly extracting and reheapifying elements, it gradually builds a sorted array from an initially unsorted one.

With its time complexity of O(n log n), heap sort is suitable for large datasets where performance is crucial.