In computer science, a priority queue is an abstract data-type similar to a regular queue or stack data structure. Each element in a priority queue has an associated priority. In a priority queue, elements with high priority are served before elements with low priority. In some implementations, if two elements have the same priority, they are served in the same order that they were enqueued in. In other implementations, the order of elements with the same priority is undefined. While priority queues are often implemented using heaps, they are conceptually distinct from heaps. A priority queue is an abstract data structure like a list or a map; just as a list can be implemented with a linked list or with an array, a priority queue can be implemented with a heap or another method such as an unordered array. Operations A priority queue must at least support the following operations: is_empty: check whether the queue has no elements. insert_with_priority: add an element to the queue with an associated priority. pull_highest_priority_element: remove the element from the queue that has the highest priority, and return it. This is also known as "pop_element(Off)", "get_maximum_element" or "get_front(most)_element". Some conventions reverse the order of priorities, considering lower values to be higher priority, so this may also be known as "get_minimum_element", and is often referred to as "get-min" in the literature. This may instead be specified as separate "peek_at_highest_priority_element" and "delete_element" functions, which can be combined to produce "pull_highest_priority_element". In addition, peek (in this context often called find-max or find-min), which returns the highest-priority element but does not modify the queue, is very frequently implemented, and nearly always executes in O(1) time. This operation and its O(1) performance is crucial to many applications of priority queues. More advanced implementations may support more complicated operations, such as pull_lowest_priority_element, inspecting the first few highest- or lowest-priority elements, clearing the queue, clearing subsets of the queue, performing a batch insert, merging two or more queues into one, incrementing priority of any element, etc. Stacks and queues can be implemented as particular kinds of priority queues, with the priority determined by the order in which the elements are inserted. In a stack, the priority of each inserted element is monotonically increasing; thus, the last element inserted is always the first retrieved. In a queue, the priority of each inserted element is monotonically decreasing; thus, the first element inserted is always