Based on a typical [heap](../tree/heap.md) data structure, Heap Sort is an [in-place, not stable comparison-baed sorting](overview.md) algorithm and can be thought of as a improved version of [selection sort](selection-sort.md) algorithm.

Initially, a [BUILD-MAX-HEAP](../tree/heap.md) is called upon a randomly distributed inputs to setup a [max-heap](../tree/heap.md). Now that the topmost root key must be the largest among the heap, extract that key to a sorted array and restore the heap property using [MAX-HEAPIFY](../tree/heap.md) operation. Repeat the procedures of extraction and adjustment till the heap is empty, the sorted array is formed.

In each heap restoration process, instead of [selection sort](selection-sort.md) method of locating the maximum or minimum key within the inputs in linear time, Heap Sort uses [MAX-HEAPIFY](../tree/heap.md) operation to reduce time complexity to logarithmic bound.

The following code transforms a randomly distributed array into an ascending entry array.

Given a n-size inputs, there is a Ο(n) estimate on [BUILD_MAX_HEAP](../tree/heap.md) operation; And for each input entry, a [MAX_HEAPIFY](../tree/heap.md) is expected to perform and thus costs Ο(n ⋅ log(n)) in total.

Since the [Heap Sort](#heap-sort) is also a [comparison-based sorting](overview.md) that has proven with a lower bound Ω(n ⋅ log(n)), the more tighter bound for Heap Sort will be Θ(n ⋅ log(n)).