Heap and Priority Queue Overview

Complete Binary Tree Representation

• A heap can be represented as a complete binary tree.
• In this representation, nodes can be efficiently managed using an array.
• The structure and index management allows us to compute the position of children and parents using formulas.

Indexing in Heaps

• In our implementation, the heap starts from index 0 rather than index 1.
• Important functions are designed based on this zero-indexing method.
• Starting from index 0 will affect calculations when seeking child or parent indices.

Helper Function Limitations

• The helper functions calculate indices but do not validate their validity in the binary tree structure.
• Additional validations are necessary to ensure indices lie within the bounds of the current size of the heap.

Heapify Operation (KDPY Function)

• The KDPY function is utilized to correct the heap properties at a specific node index.
• It compares the node with its children and swaps values if necessary, continuing until the heap property is restored (the node is less than its children).
• Stopping Conditions:
• No swapping occurs (the node is the smallest).
• The node is a leaf (no children to compare).

Example of Heapify Process:

• If the value 91 needs to be fixed, it checks the values at its position and its children.
• Swaps are made with the smallest child until the heap property is satisfied.

Complexity of Heapify

• The worst-case time complexity for the heapify operation is O(log n) due to traversal along the height of the tree.

Building a Heap from an Array

• To convert an arbitrary array into a heap, start from the last internal node and move towards the root, calling heapify on each node.
• Leaf nodes are skipped since they inherently satisfy heap properties.
• The final complexity for building the heap is refined to O(n) through a summation approach, contrary to the naive O(n log n).

Applications of Heaps

• Heap Sort:

• Consists of pulling the minimum element and re-heapifying until sorted.

• Heap property ensures that the minimum element remains at the root, facilitating efficient sorting.

• The final output requires an in-place adjustment to ensure sorted order.

• Priority Queues:

• Heaps support the priority queue structure by maintaining order based on priority rather than enqueue order.

• Implementation of insert, decrease key, extract min functions allow efficient management of priorities.

Heap Sort Algorithm Steps:

• Build a min-heap from the array.
• Extract the minimum element (root), place it at the end of the array (shrink heap size).
• Call heapify to restore heap properties, then repeat until the array is sorted.

Insertion in Priority Queue:

• New elements are added to the end of the heap (array) and may necessitate an upward adjustment via a heapify-up operation potential violation of heap property.

Code Considerations:

• Special cases such as heap being empty or single element must be managed explicitly in the code implementations for extraction and minimum retrieval.
• Efficiency in space and operations must be considered in understanding heap behavior and performance.