heapq operates directly on a Python list, treating it as a binary heap with heap[0] as the smallest element. heappush and heappop are both O(log n). Note that the heap is NOT sorted: it satisfies the heap invariant (parent <= children), but the in-memory order can look scrambled. Reading heap[0] gives an O(1) min query, which is the headline operation. To sort, repeatedly pop until empty (heap sort), which is O(n log n) total.

heapq.nlargest(k, iter) and heapq.nsmallest(k, iter) return the K largest or smallest elements in O(n log k) time, which is asymptotically better than sorted(...)[:k] (O(n log n)) when k is much smaller than n. They both accept an optional key argument for projection, mirroring sorted. For very small K (1, 2) the constant factors favour min/max instead, but for K up to a few hundred the heap approach is the right default.

heapq.merge(*iters) lazily merges any number of pre-sorted iterables into one sorted stream. It uses a heap of size K (the number of inputs) so memory is bounded regardless of how long each input is, and total time is O(N log K) for N total items. This is the streaming-friendly answer to 'merge K sorted lists' and the building block for external sorts on data too big for RAM. The optional key argument lets you merge dicts or objects by an extracted field.