Equivalence Between Priority Queues and Sorting

Keywords and Synonyms

Heap          

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 [4,7].

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 ${ (Q_{1}, Q_{2}) }$::

Given priority queues Q ₁ and Q ₂, 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 meldas...