Heap: Structure and Applications, Introduction to Min-Heap and Max-Heap
A heap is a special type of complete binary tree, characterized by the size relationship between parent and child nodes (parent ≤ child for a min-heap, parent ≥ child for a max-heap). It efficiently retrieves extreme values (with the top element being the minimum or maximum), similar to a priority queue. The underlying structure is a complete binary tree, where each level is filled as much as possible, and the last level is filled from left to right. When stored in an array, the left child index is 2i+1, the right child index is 2i+2, and the parent index is (i-1)//2. Basic operations include insertion (appending to the end and then "bubbling up") and deletion (replacing the top element with the last element and then "bubbling down"), both with a time complexity of O(log n). Heaps are widely used in priority queues (e.g., task scheduling), finding the k-th largest element, and Huffman coding. They are a critical structure for efficiently handling extreme value problems.
Read MoreImplementing Heap Sort Algorithm with Python
Heap Sort is an efficient sorting algorithm that leverages the heap data structure, with a stable time complexity of O(n log n) and a space complexity of O(1), making it suitable for sorting large-scale data. A heap is a complete binary tree where parent node values are either greater than or equal to (max heap) or less than or equal to (min heap) their child node values. In an array representation, the indices of a heap follow these relationships: the children of a parent node at index i are located at 2i+1 and 2i+2, while the parent of a child node at index j is at (j-1)//2. The core operations include: 1. **Heapify**: Adjust the subtree rooted at index i to be a max heap by recursively comparing child nodes and swapping values as needed. 2. **Build Max Heap**: Starting from the last non-leaf node (n//2 - 1) and moving upward, adjust all nodes to ensure the entire tree satisfies the max heap property. The sorting process involves: first building the max heap, then repeatedly swapping the root (maximum value) with the last element of the heap, followed by calling Heapify to re-adjust the remaining elements into a max heap. This results in a sorted array from smallest to largest. Heap Sort is an in-place sorting algorithm, making it well-suited for scenarios with large data volumes.
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