Priority queue

In computer science, a priority queue is an abstract data type which is like a regular queue or stack data structure, but where additionally each element has a "priority" associated with it. In a priority queue, an element with high priority is served before an element with low priority. If two elements have the same priority, they are served according to their order in the queue.

• stack — elements are pulled in last-in first-out-order (e.g. a stack of papers)
• queue — elements are pulled in first-in first-out-order (e.g. a line in a cafeteria)

It is a common misconception that a priority queue is a heap. A priority queue is an abstract concept like "a list" or "a map"; just as a list can be implemented with a linked list or an array, a priority queue can be implemented with a heap or a variety of other methods.

A priority queue must at least support the following operations:

• 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 (also known as "pop_element(Off)", "get_maximum_element", or "get_front(most)_element"; some conventions consider lower priorities to be higher, so this may also be known as "get_minimum_element", and is often referred to as "get-min" in the literature; the literature also sometimes implement separate "peek_at_highest_priority_element" and "delete_element" functions, which can be combined to produce "pull_highest_priority_element")

More advanced implementations may support more complicated operations, such as pull_lowest_priority_element, inspecting the first few highest- or lowest-priority elements (peeking at the highest priority element can be made O(1) time in nearly all implementations), 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.

Similarity to queues

One can imagine a priority queue as a modified queue, but when one would get the next element off the queue, the highest-priority element is retrieved first.

Stacks and queues may be modeled as particular kinds of priority queues. 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 the first retrieved.

Implementation

Naive implementations

There are a variety of simple, usually inefficient, ways to implement a priority queue. They provide an analogy to help one understand what a priority queue is. For instance, one can keep all the elements in an unsorted list. Whenever the highest-priority element is requested, search through all elements for the one with the highest priority. (In big O notation: O(1) insertion time, O(n) pull time due to search.)

Usual implementation

To improve performance, priority queues typically use a heap as their backbone, giving O(log n) performance for inserts and removals, and O(n) to build initially. Alternatively, when a self-balancing binary search tree is used, insertion and removal also take O(log n) time, although building trees from existing sequences of elements takes O(n log n) time; this typical where one might already have access to these data structures, such as with third-party or standard libraries.

Note that from a computational-complexity standpoint, priority queues are congruent to sorting algorithms. See the next section for how efficient sorting algorithms can create efficient priority queues.

There are several specialized heap data structures that either supply additional operations or outperform these approaches. The binary heap uses O(log n) time for both operations, but also allow queries of the element of highest priority without removing it in constant time. Binomial heaps add several more operations, but require O(log n) time for requests. Fibonacci heaps can insert elements, query the highest priority element, and increase an element's priority in amortized constant time though deletions are still O(log n)). Brodal queues can do this in worst-case constant time.

While relying on a heap is a common way to implement priority queues, for integer data, faster implementations exist. This can even apply to data-types that have a finite range, such as floats:

• When the set of keys is {1, 2, ..., C}, a van Emde Boas tree would support the minimum, maximum, insert, delete, search, extract-min, extract-max, predecessor and successor operations in $O(\log \log C)$ time, but has a space cost for small queues of about O(2^m/2), where m is the number of bits in the priority value.
• The Fusion tree algorithm by Fredman and Willard implements the minimum operation in O(1) time and insert and extract-min operations in $O({\sqrt {\log n}})$ time.

For applications that do many "peek" operations for every "extract-min" operation, the time complexity for peek actions can be reduced to O(1) in all tree and heap implementations by caching the highest priority element after every insertion and removal. For insertion, this adds at most a constant cost, since the newly inserted element is compared only to the previously cached minimum element. For deletion, this at most adds an additional "peek" cost, which is typically cheaper than the deletion cost, so overall time complexity is not significantly impacted.

Equivalence of priority queues and sorting algorithms

Using a priority queue to sort

The semantics of priority queues naturally suggest a sorting method: insert all the elements to be sorted into a priority queue, and sequentially remove them; they will come out in sorted order. This is actually the procedure used by several sorting algorithms, once the layer of abstraction provided by the priority queue is removed. This sorting method is equivalent to the following sorting algorithms:

• Heapsort if the priority queue is implemented with a heap.
• Smoothsort if the priority queue is implemented with a Leonardo heap.
• Selection sort if the priority queue is implemented with an unordered array.
• Insertion sort if the priority queue is implemented with an ordered array.
• Tree sort if the priority queue is implemented with a self-balancing binary search tree.

Using a sorting algorithm to make a priority queue

A sorting algorithm can also be used to implement a priority queue. Specifically, Thorup says:

We present a general deterministic linear space reduction from priority queues to sorting implying that if we can sort up to n keys in S(n) time per key, then there is a priority queue supporting delete and insert in O(S(n)) time and find-min in constant time.

That is, if there is a sorting algorithm which can sort in O(S) time per key, where S is some function of n and word size , then one can use the given procedure to create a priority queue where pulling the highest-priority element is O(1) time, and inserting new elements (and deleting elements) is O(S) time. For example if one has an O(n lg(lg(n))) sort algorithm, one can easily create a priority queue with O(1) pulling and O(lg(lg(n))) insertion.

Libraries

A priority queue is often considered to be a "container data structure".

The Standard Template Library (STL), and the C++ 1998 standard, specifies priority_queue as one of the STL container adaptor class templates. It implements a max-priority-queue. Unlike actual STL containers, it does not allow iteration of its elements (it strictly adheres to its abstract data type definition). STL also has utility functions for manipulating another random-access container as a binary max-heap.

Python's heapq module implements a binary min-heap on top of a list.

Java's library contains a PriorityQueue class, which implements a min-priority-queue.

Go's library contains a container/heap module, which implements a min-heap on top of any compatible data structure.

The Standard PHP Library extension contains the class SplPriorityQueue.

Apple's Core Foundation framework contains a CFBinaryHeap structure, which implements a min-heap.

Applications

Bandwidth management

Priority queuing can be used to manage limited resources such as bandwidth on a transmission line from a network router. In the event of outgoing traffic queuing due to insufficient bandwidth, all other queues can be halted to send the traffic from the highest priority queue upon arrival. This ensures that the prioritized traffic (such as real-time traffic, e.g. an RTP stream of a VoIP connection) is forwarded with the least delay and the least likelihood of being rejected due to a queue reaching its maximum capacity. All other traffic can be handled when the highest priority queue is empty. Another approach used is to send disproportionately more traffic from higher priority queues.

Many modern protocols for Local Area Networks also include the concept of Priority Queues at the Media Access Control (MAC) sub-layer to ensure that high-priority applications (such as VoIP or IPTV) experience lower latency than other applications which can be served with Best effort service. Examples include IEEE 802.11e (an amendment to IEEE 802.11 which provides Quality of Service) and ITU-T G.hn (a standard for high-speed Local area network using existing home wiring (power lines, phone lines and coaxial cables).

Usually a limitation (policer) is set to limit the bandwidth that traffic from the highest priority queue can take, in order to prevent high priority packets from choking off all other traffic. This limit is usually never reached due to high level control instances such as the Cisco Callmanager, which can be programmed to inhibit calls which would exceed the programmed bandwidth limit.

Discrete event simulation

Another use of a priority queue is to manage the events in a discrete event simulation. The events are added to the queue with their simulation time used as the priority. The execution of the simulation proceeds by repeatedly pulling the top of the queue and executing the event thereon.

See also: Scheduling (computing), queueing theory

Dijkstra's algorithm

When the graph is stored in the form of adjacency list or matrix, priority queue can be used to extract minimum efficiently when implementing Dijkstra's algorithm, although one also needs the ability to alter the priority of a particular vertex in the priority queue efficiently.

Huffman coding

Huffman coding requires one to repeatedly obtain the two lowest-frequency trees. A priority queue makes this efficient.

A* and SMA* search algorithms

The A* search algorithm finds the shortest path between two vertices or nodes of a weighted graph, trying out the most promising routes first. The priority queue (also known as the fringe) is used to keep track of unexplored routes; the one for which a lower bound on the total path length is smallest is given highest priority. If memory limitations make A* impractical, the SMA* algorithm can be used instead, with a double-ended priority queue to allow removal of low-priority items.

ROAM triangulation algorithm

The Real-time Optimally Adapting Meshes (ROAM) algorithm computes a dynamically changing triangulation of a terrain. It works by splitting triangles where more detail is needed and merging them where less detail is needed. The algorithm assigns each triangle in the terrain a priority, usually related to the error decrease if that triangle would be split. The algorithm uses two priority queues, one for triangles that can be split and another for triangles that can be merged. In each step the triangle from the split queue with the highest priority is split, or the triangle from the merge queue with the lowest priority is merged with its neighbours.

See also

• Batch queue
• Closure
• Command pattern
• Command queue
• Function object
• Job scheduler
• Model-view-controller

Further reading

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 6.5: Priority queues, pp.138–142.

External links

• C++ reference for std::priority_queue
• Descriptions by Lee Killough
• PQlib - Open source Priority Queue library for C
• libpqueue is a generic priority queue (heap) implementation (in C) used by the Apache HTTP Server project.
• Survey of known priority queue structures by Stefan Xenos
• UC Berkeley - Computer Science 61B - Lecture 24: Priority Queues (video) - introduction to priority queues using binary heap
• Double-Ended Priority Queues by Sartaj Sahni