Heapify

🔥 Priority Queue → Heap-based priority_queue implementations rely on heapify (sift-down/up) to maintain the heap invariant during insert and extract operations and to build the heap in O(n).

🔥 Heap Sort → Heapsort relies on heapify to build the initial heap in O(n) and to restore the heap property after each root extraction, enabling its in-place O(n log n) sorting procedure.