Heap Sort: How to Implement Heap Sort and Detailed Explanation of Time Complexity

Heap sort is a sorting algorithm that utilizes "heaps" (a special type of complete binary tree), commonly using a max heap (where parent nodes are greater than or equal to their child nodes). The core idea is "build the heap first, then sort": first convert the array into a max heap (with the maximum value at the heap top), then repeatedly swap the heap top with the last element, adjust the remaining elements into a heap, and complete the sorting. Basic concepts of heaps: A complete binary tree structure where for an element at index i in the array, the left child is at 2i+1, the right child at 2i+2, and the parent is at (i-1)//2. In a max heap, parent nodes are greater than or equal to their children; in a min heap, parent nodes are less than or equal to their children. The implementation has two main steps: 1. Constructing the max heap: Starting from the last non-leaf node, use "heapify" (comparing parent and child nodes, swapping the maximum value, and recursively adjusting the subtree) to ensure the max heap property is maintained. 2. Sorting: Swap the heap top with the last unsorted element, reduce the heap size, and repeat the heapify process until sorting is complete. Time complexity: Building the heap takes O(n), and the sorting process takes O(n log n), resulting in an overall time complexity of O(n log n). Space complexity is O(1) (in-place sorting). It is an unstable sort and suitable for sorting large-scale data.