Implementing and using priority queues, often with a heap.
A Priority Queue is an abstract data type similar to a regular queue, but with an important difference: each element has an associated 'priority'. Elements with higher priority are served before elements with lower priority. If two elements have the same priority, their relative order is typically based on their order in the queue. The most common and efficient way to implement a priority queue is by using a heap data structure. A heap is a specialized tree-based data structure that satisfies the heap property: in a max-heap, for any given node `C`, if `P` is a parent node of `C`, then the key (the value) of `P` is greater than or equal to the key of `C`. This ensures that the element with the highest priority (maximum value) is always at the root of the tree, allowing for O(1) access to it. Inserting a new element (`push`) and extracting the max element (`pop`) both take O(log n) time, as they may require adjustments to maintain the heap property. Priority queues are incredibly useful in many algorithms, including Dijkstra's shortest path algorithm, Prim's algorithm for minimum spanning trees, and Huffman coding for data compression.