Equivalence Between Priority Queues and Sorting

Synonyms

Heap

Years and Authors of Summarized Original Work

• 2007 (2002); Thorup

Problem Definition

A priority queue is an abstract data structure that maintains a set Q of elements, each with an associated value called a key, under the following set of operations [5, 6]:

insert( Q, x, k ): Inserts element x with key k into Q.

find-min( Q ): Returns an element of Q with the minimum key but does not change Q.

delete( Q, x, k ): Deletes element x with key k from Q.

Additionally, the following operations are often supported:

delete-min( Q ): Deletes an element with the minimum key value from Q and returns it.

decrease-key( Q, x, k ): Decreases the current key $k^{{\prime}}$ of x to k assuming $k < k^{{\prime}}$.

meld( Q₁ ,Q₂ ): Given priority queues Q₁ and Q₂, returns the priority queue $Q_{1} \cup Q_{2}$.

Observe that a delete-min can be implemented as a find-min followed by a delete, a decrease-key as a delete followed by an insert, and a meld as a series of find-min, delete and insert...