How does the heapsort algorithm sort numbers?

Heapsort algorithm sorts numbers by building a binary heap and then removing the maximum element until the heap is empty.

Heapsort is a comparison-based sorting algorithm that operates on an array of numbers. It uses a data structure known as a binary heap, which is a complete binary tree that satisfies the heap property. This property requires that 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.

The heapsort algorithm begins by transforming the input array into a max heap. This is done by iterating over the array from the middle to the beginning, and for each element, 'sifting it down' to its correct position in the heap. This process ensures that the largest element is at the root of the heap.

Once the heap is constructed, the algorithm repeatedly swaps the root (the maximum element) with the last element of the heap, effectively placing the maximum element in its final sorted position in the array. The size of the heap is then reduced by one (excluding the last element from the heap), and the new root is sifted down to its correct position. This process is repeated until the heap is empty, resulting in a sorted array.

The beauty of heapsort lies in its efficiency. It has a worst-case and average time complexity of O(n log n), making it one of the most efficient sorting algorithms for large data sets. Moreover, it performs sorting in-place, requiring only a constant amount of additional space.

However, heapsort is not a stable sort; that is, equal elements may not retain their relative order in the sorted array. This is because heapsort moves elements to their final position without considering their initial positions. Despite this, heapsort is widely used due to its optimal time complexity and minimal space requirement.