Data Structures and Algorithms
with Object-Oriented Design Patterns in C# |

A priority queue is a container which provides the following three operations:

`Enqueue`- used to put objects into the container;
`Min`- accesses the smallest object in the container; and
`DequeueMin`-
removes the smallest object from the container.

Program defines the `PriorityQueue` interface.
The `PriorityQueue` interface extends the `Container` interface
defined in Program .
In addition to the inherited methods,
the `PriorityQueue` interface comprises
the three methods listed above.

**Program:** `PriorityQueue` interface.

Program defines the
`MergeablePriorityQueue` interface.
The `MergeablePriorityQueue` interface extends
the `PriorityQueue` interface defined in Program .
A *mergeable priority queue*
is one which provides the ability to
merge efficiently two priority queues into one.
Of course it is always possible to merge two priority queues
by dequeuing the elements of one queue
and enqueueing them in the other.
However, the mergeable priority queue implementations we will consider
allow more efficient merging than this.

**Program:** `MergeablePriorityQueue` interface.

It is possible to implement the required functionality
using data structures that we have already considered.
For example, a priority queue can be implemented simply as a list.
If an *unsorted list* is used,
enqueueing can be accomplished in constant time.
However, finding the minimum and removing the minimum each require *O*(*n*) time
where *n* is the number of items in the queue.
On the other hand,
if a *sorted list* is used,
finding the minimum and removing it is easy--both operations can be done in constant time.
However, enqueueing an item in a sorted list requires *O*(*n*) time.

Another possibility is to use a search tree.
For example, if an *AVL tree* is used
to implement a priority queue,
then all three operations can be done in time.
However, search trees provide more functionality than we need.
Search trees support finding the largest item with `Max`,
deletion of arbitrary objects with `Withdraw`,
and the ability to visit in order all the contained objects
via `DepthFirstTraversal`.
All these operations can be done as efficiently
as the priority queue operations.
Because search trees support more methods
than we really need for priority queues,
it is reasonable to suspect that there are more efficient ways
to implement priority queues.
And indeed there are!

A number of different priority queue implementations
are described in this chapter.
All the implementations have one thing in common--they are all based on a special kind of tree called a *min heap*
or simply a *heap*.

According to Definition , the key in each node of a heap is less than or equal to the roots of all the subtrees of that node. Therefore, by induction, the key in each node is less than or equal to all the keys contained in the subtrees of that node. Note, however, that the definition says nothing about the relative ordering of the keys in the subtrees of a given node. For example, in a binary heap either the left or the right subtree of a given node may have the larger key.Definition ((Min) Heap)A(Min) Heapis a tree,

with the following properties:

- Every subtree of
Tis a heap; and,- The root of
Tis less than or equal to the root of every subtree ofT. That is, for alli, , where is the root of .

Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.