Heapsort and Priority Queues

"Give me the current top 10 holdings by weight" and "process the next order in price-time priority" are both instances of the same computational primitive: repeatedly extract the largest (or smallest) element from a changing collection, without re-sorting the whole thing each time. The binary heap is the data structure built exactly for this, and heapsort is the $\Theta(n\log n)$, in-place sorting algorithm that falls out of it almost for free. This note builds the heap from its array representation up, derives the running times of heapify, build-heap, and heapsort, and introduces the priority queue ADT the heap supports — the structure Python's heapq module implements directly.