Double-Ended Priority Queues:Definition and an Application.

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Definition and an Application

A double-ended priority queue (DEPQ) is a collection of zero or more elements. Each element has a priority or value. The operations performed on a double-ended priority queue are:

1. getM in() ... return element with minimum priority

2. getM ax() ... return element with maximum priority

3. put(x) ... insert the element x into the DEPQ

4. removeM in() ... remove an element with minimum priority and return this element

5. removeM ax() ... remove an element with maximum priority and return this element

One application of a DEPQ is to the adaptation of quick sort, which has the the best expected run time of all known internal sorting methods, to external sorting. The basic idea in quick sort is to partition the elements to be sorted into three groups L, M , and R. The middle group M contains a single element called the pivot, all elements in the left group L are ≤ the pivot, and all elements in the right group R are ≥ the pivot. Following this partitioning, the left and right element groups are sorted recursively.

In an external sort, we have more elements than can be held in the memory of our computer. The elements to be sorted are initially on a disk and the sorted sequence is to be left on the disk. When the internal quick sort method outlined above is extended to an external quick sort, the middle group M is made as large as possible through the use of a DEPQ. The external quick sort strategy is:

1. Read in as many elements as will fit into an internal DEPQ. The elements in the DEPQ will eventually be the middle group of elements.

2. Read in the remaining elements. If the next element is ≤ the smallest element in the DEPQ, output this next element as part of the left group. If the next element is ≥ the largest element in the DEPQ, output this next element as part of the right group. Otherwise, remove either the max or min element from the DEPQ (the choice may be made randomly or alternately); if the max element is removed, output it as part of the right group; otherwise, output the removed element as part of the left group; insert the newly input element into the DEPQ.

3. Output the elements in the DEPQ, in sorted order, as the middle group.

4. Sort the left and right groups recursively.

In this chapter, we describe four implicit data structures—symmetric min-max heaps, interval heaps, min-max heaps, and deaps—for DEPQs. Also, we describe generic methods to obtain efficient DEPQ data structures from efficient data structures for single-ended priority queues (PQ).1

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