Heap Operations

Heaps are crucial in data structures, providing efficient ways to manage and organize data. Key operations like heapify, insert, and delete help maintain the heap property, making heaps ideal for priority queues and sorting algorithms.

1. Heapify

   • Transforms an unordered array into a valid heap structure.
   • Ensures the heap property is maintained, where each parent node is greater (max-heap) or less (min-heap) than its children.
   • Typically implemented using a bottom-up approach, starting from the last non-leaf node.
   • Time complexity is O(n) for building a heap from an array.

2. Insert

   • Adds a new element to the heap while maintaining the heap property.
   • Involves placing the new element at the end of the heap and "bubbling up" to restore order.
   • Time complexity is O(log n) due to the potential height of the heap.
   • Important for dynamic data structures where elements are frequently added.

3. Delete (Extract) Max/Min

   • Removes the root element (maximum in max-heap or minimum in min-heap) and returns it.
   • Replaces the root with the last element in the heap and "bubbles down" to restore the heap property.
   • Time complexity is O(log n) as it may require traversing the height of the heap.
   • Essential for priority queue operations.

4. Build Heap

   • A specific process to create a heap from an unsorted array.
   • Utilizes the heapify process iteratively for all non-leaf nodes.
   • Efficiently organizes data for subsequent heap operations.
   • Time complexity is O(n), making it faster than inserting elements one by one.

5. Heap Sort

   • A comparison-based sorting algorithm that uses the heap data structure.
   • Involves building a heap from the input data and repeatedly extracting the maximum/minimum.
   • Time complexity is O(n log n), making it efficient for large datasets.
   • Not stable, meaning the relative order of equal elements may not be preserved.

6. Increase/Decrease Key

   • Modifies the value of a specific element in the heap and adjusts the structure to maintain the heap property.
   • For increasing a key, the element is "bubbled up"; for decreasing, it is "bubbled down."
   • Time complexity is O(log n) for both operations.
   • Useful in algorithms like Dijkstra's for priority queue updates.

7. Merge Heaps

   • Combines two heaps into a single heap while maintaining the heap property.
   • Can be done in various ways, including creating a new heap or modifying one of the existing heaps.
   • Time complexity can vary; a naive approach is O(n + m), where n and m are the sizes of the heaps.
   • Important for applications that require dynamic merging of priority queues.