Heap Data Structure

In algorithms, a heap is a specialized binary tree-based data structure that satisfies the heap property:

each parent node is either greater than or equal to (max-heap) or less than or equal to (min-heap) its children.

Locally ordered.

Key Characteristics

Feature	Description
Structure	Typically implemented as a binary heap stored in an array.
Use Cases	Priority queues, heapsort, scheduling, Dijkstra’s and Prim’s algorithms.
Efficiency	insert() and extract_min() / extract_max() run in $O(\log n)$ time.

Why Use a Heap?

Efficient Priority Queue

• Provides constant-time access to the highest- or lowest-priority element.
• Suitable when priorities change over time or are not known up front.
• Use cases: task schedulers, event-driven simulation, packet routing.

Repeated Min/Max Extraction

• Ideal for problems that require frequent retrieval and removal of extremal elements while maintaining partial order.
• Time complexity:
  • insert(): $O(\log n)$
  • extract_min() / extract_max(): $O(\log n)$
  • peek_min() / peek_max(): $O(1)$

When to Use a Heap

Use a heap when your problem involves:

• Maintaining a dynamic priority queue.
• Fast access to the smallest or largest element.
• Efficient partial ordering without full sorting.
• Managing a sliding or streaming window of values (e.g. top-k, median).

Python Example (Min-Heap)

import heapq
 
heap = []
heapq.heappush(heap, 3)
heapq.heappush(heap, 1)
heapq.heappush(heap, 2)
 
print(heapq.heappop(heap))  # Output: 1