Binary Heaps and Heap Operations

Binary Heaps

• Binary heaps are a data structure for implementing priority queues efficiently.
• Based on complete binary trees, which are almost perfectly balanced.
• Height of a complete tree with n nodes is approximately log₂(n).
• Stores data in an array, representing the tree in level order.
• Nodes contain a key.
• Heap order property: Each parent's key is greater than or equal to its children's keys. This means the largest key is always at the root (index 1 in the array).
• No explicit links are needed for navigating the tree; arithmetic operations on indices are used instead.

Heap Operations

• Parent of node at index k: k/2

• Children of node at index k: 2k and 2k + 1

• Swim operation: Used when a key value increases. Restores the heap property when a value is increased in a node.

• Sink operation: Used when a key value decreases. Restores the heap property when a value is decreased in any node.

• These operations allow for efficient priority queue operations (insert and delete) in O(log n) time.

Significance of Binary Heaps

• Simple structure.
• Efficient data representation.
• Logarithmic time complexity for crucial operations.