Lecture 323:- Introduction to Heap

A heap is a specialized tree-based data structure that satisfies the heap property. Heaps are typically used to efficiently find and extract the maximum or minimum element from a collection of elements. They are commonly implemented as binary trees, specifically binary heaps.

Binary Heaps: Binary heaps are binary trees that adhere to the heap property. There are two types of binary heaps:

1. Max Heap: In a max heap, for any given node, the value of the node is greater than or equal to the values of its children. Thus, the maximum element in the heap is always at the root.
2. Min Heap: In a min heap, for any given node, the value of the node is less than or equal to the values of its children. Therefore, the minimum element in the heap is always at the root.

Heap Property: The heap property ensures that the structure maintains its order, allowing efficient retrieval of the maximum or minimum element. In a max heap, the maximum element is at the root, and for any node in the heap, its value is greater than or equal to its children. In a min heap, the minimum element is at the root, and for any node in the heap, its value is less than or equal to its children.

Common Operations on Heap:

1. Insertion: Adding a new element to the heap while maintaining the heap property.
2. Extract Maximum/Minimum: Removing and returning the maximum or minimum element from the heap, respectively.
3. Peek: Viewing the maximum or minimum element without removing it from the heap.
4. Heapify: Converting an array of elements into a heap.

Applications of Heaps: Heaps are widely used in various algorithms and data structures, including:

• Priority Queues: Heaps are used to implement priority queues, where elements are retrieved based on their priority.
• Heap Sort: Heap sort is an efficient sorting algorithm that uses a binary heap to sort elements.
• Dijkstra's Shortest Path Algorithm: Heaps are used to efficiently find the minimum distance in graph traversal algorithms like Dijkstra's algorithm.
• Huffman Coding: Heaps are used in data compression algorithms like Huffman coding to encode and decode data efficiently.

Heaps provide efficient access to the maximum or minimum element, making them a valuable tool in various computer science applications. Their binary tree structure and the heap property enable them to perform essential operations with logarithmic time complexity, making them suitable for large datasets and time-critical applications.