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In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (inclu ...
, a queue is a collection of entities that are maintained in a sequence and can be modified by the addition of entities at one end of the sequence and the removal of entities from the other end of the sequence. By convention, the end of the sequence at which elements are added is called the back, tail, or rear of the queue, and the end at which elements are removed is called the head or front of the queue, analogously to the words used when people line up to wait for goods or services. The operation of adding an element to the rear of the queue is known as ''enqueue'', and the operation of removing an element from the front is known as ''dequeue''. Other operations may also be allowed, often including a '' peek'' or ''front'' operation that returns the value of the next element to be dequeued without dequeuing it. The operations of a queue make it a first-in-first-out (FIFO) data structure. In a FIFO data structure, the first element added to the queue will be the first one to be removed. This is equivalent to the requirement that once a new element is added, all elements that were added before have to be removed before the new element can be removed. A queue is an example of a linear data structure, or more abstractly a sequential collection. Queues are common in computer programs, where they are implemented as data structures coupled with access routines, as an abstract data structure or in object-oriented languages as classes. Common implementations are circular buffers and
linked list In computer science, a linked list is a linear collection of data elements whose order is not given by their physical placement in memory. Instead, each element points to the next. It is a data structure consisting of a collection of nodes whic ...
s. Queues provide services in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (inclu ...
,
transport Transport (in British English), or transportation (in American English), is the intentional movement of humans, animals, and goods from one location to another. Modes of transport include air, land ( rail and road), water, cable, pipe ...
, and
operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
where various entities such as data, objects, persons, or events are stored and held to be processed later. In these contexts, the queue performs the function of a buffer. Another usage of queues is in the implementation of
breadth-first search Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next d ...
.

# Queue implementation

Theoretically, one characteristic of a queue is that it does not have a specific capacity. Regardless of how many elements are already contained, a new element can always be added. It can also be empty, at which point removing an element will be impossible until a new element has been added again. Fixed-length arrays are limited in capacity, but it is not true that items need to be copied towards the head of the queue. The simple trick of turning the array into a closed circle and letting the head and tail drift around endlessly in that circle makes it unnecessary to ever move items stored in the array. If n is the size of the array, then computing indices modulo n will turn the
array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
into a circle. This is still the conceptually simplest way to construct a queue in a high-level language, but it does admittedly slow things down a little, because the array indices must be compared to zero and the array size, which is comparable to the time taken to check whether an array index is out of bounds, which some languages do, but this will certainly be the method of choice for a quick and dirty implementation, or for any high-level language that does not have pointer syntax. The array size must be declared ahead of time, but some implementations simply double the declared array size when overflow occurs. Most modern languages with objects or pointers can implement or come with libraries for dynamic lists. Such
data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships among them, ...
s may have not specified a fixed capacity limit besides memory constraints. Queue ''overflow'' results from trying to add an element onto a full queue and queue ''underflow'' happens when trying to remove an element from an empty queue. A ''bounded queue'' is a queue limited to a fixed number of items. There are several efficient implementations of FIFO queues. An efficient implementation is one that can perform the operations—en-queuing and de-queuing—in O(1) time. *
Linked list In computer science, a linked list is a linear collection of data elements whose order is not given by their physical placement in memory. Instead, each element points to the next. It is a data structure consisting of a collection of nodes whic ...
**A doubly linked list has O(1) insertion and deletion at both ends, so it is a natural choice for queues. **A regular singly linked list only has efficient insertion and deletion at one end. However, a small modification—keeping a pointer to the ''last'' node in addition to the first one—will enable it to implement an efficient queue. *A deque implemented using a modified dynamic array

## Queues and programming languages

Queues may be implemented as a separate data type, or maybe considered a special case of a double-ended queue (deque) and not implemented separately. For example,
Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offi ...
and
Ruby A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum ( aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called ...
allow pushing and popping an array from both ends, so one can use push and shift functions to enqueue and dequeue a list (or, in reverse, one can use unshift and pop), although in some cases these operations are not efficient. C++'s Standard Template Library provides a "queue" templated class which is restricted to only push/pop operations. Since J2SE5.0, Java's library contains a interface that specifies queue operations; implementing classes include and (since J2SE 1.6) . PHP has a
SplQueue
class and third party libraries like beanstalk'd and Gearman.

## Example

A simple queue implemented in
JavaScript JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, of ...
: class Queue

# Purely functional implementation

Queues can also be implemented as a
purely functional data structure In computer science, a purely functional data structure is a data structure that can be implemented in a purely functional language. The main difference between an arbitrary data structure and a purely functional one is that the latter is (strongl ...
. There are two implementations. The first one only achieves $O\left(1\right)$ per operation ''on average''. That is, the
amortized In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case ...
time is $O\left(1\right)$, but individual operations can take $O\left(n\right)$ where ''n'' is the number of elements in the queue. The second implementation is called a real-time queue and it allows the queue to be persistent with operations in O(1) worst-case time. It is a more complex implementation and requires lazy lists with memoization.

## Amortized queue

This queue's data is stored in two singly-linked lists named $f$ and $r$. The list $f$ holds the front part of the queue. The list $r$ holds the remaining elements (a.k.a., the rear of the queue) ''in reverse order''. It is easy to insert into the front of the queue by adding a node at the head of $f$. And, if $r$ is not empty, it is easy to remove from the end of the queue by removing the node at the head of $r$. When $r$ is empty, the list $f$ is reversed and assigned to $r$ and then the head of $r$ is removed. The insert ("enqueue") always takes $O\left(1\right)$ time. The removal ("dequeue") takes $O\left(1\right)$ when the list $r$ is not empty. When $r$ is empty, the reverse takes $O\left(n\right)$ where $n$ is the number of elements in $f$. But, we can say it is $O\left(1\right)$
amortized In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case ...
time, because every element in $f$ had to be inserted and we can assign a constant cost for each element in the reverse to when it was inserted.

## Real-time queue

The real-time queue achieves $O\left(1\right)$ time for all operations, without amortization. This discussion will be technical, so recall that, for $l$ a list, $, l,$ denotes its length, that ''NIL'' represents an empty list and $\operatorname\left(h,t\right)$ represents the list whose head is ''h'' and whose tail is ''t''. The data structure used to implement our queues consists of three singly-linked lists $\left(f,r,s\right)$ where ''f'' is the front of the queue, ''r'' is the rear of the queue in reverse order. The invariant of the structure is that ''s'' is the rear of ''f'' without its $, r,$ first elements, that is $, s, =, f, -, r,$. The tail of the queue $\left(\operatorname\left(x,f\right),r,s\right)$ is then almost $\left(f,r,s\right)$ and inserting an element ''x'' to $\left(f,r,s\right)$ is almost $\left(f,\operatorname\left(x,r\right),s\right)$. It is said almost, because in both of those results, $, s, =, f, -, r, +1$. An auxiliary function $aux$ must then be called for the invariant to be satisfied. Two cases must be considered, depending on whether $s$ is the empty list, in which case $, r, =, f, +1$, or not. The formal definition is $\operatorname\left(f,r,\operatorname\left(\_,s\right)\right)=\left(f,r,s\right)$ and $\operatorname\left(f,r,\text\right)=\left(f\text{'},\text,f\text{'}\right)$ where $f\text{'}$ is ''f'' followed by ''r'' reversed. Let us call $\operatorname\left(f,r\right)$ the function which returns ''f'' followed by ''r'' reversed. Let us furthermore assume that $, r, =, f, +1$, since it is the case when this function is called. More precisely, we define a lazy function $\operatorname\left(f,r,a\right)$ which takes as input three list such that $, r, =, f, +1$, and return the concatenation of ''f'', of ''r'' reversed and of ''a''. Then $\operatorname\left(f,r\right)=\operatorname\left(f,r,\text\right)$. The inductive definition of rotate is $\operatorname\left(\text,\operatorname\left(y,\text\right),a\right)=\operatorname\left(y,a\right)$ and $\operatorname\left(\operatorname\left(x,f\right),\operatorname\left(y,r\right),a\right)=\operatorname\left(x,\operatorname\left(f,r,\operatorname\left(y,a\right)\right)\right)$. Its running time is $O\left(r\right)$, but, since lazy evaluation is used, the computation is delayed until the results is forced by the computation. The list ''s'' in the data structure has two purposes. This list serves as a counter for $, f, -, r,$, indeed, $, f, =, r,$ if and only if ''s'' is the empty list. This counter allows us to ensure that the rear is never longer than the front list. Furthermore, using ''s'', which is a tail of ''f'', forces the computation of a part of the (lazy) list ''f'' during each ''tail'' and ''insert'' operation. Therefore, when $, f, =, r,$, the list ''f'' is totally forced. If it was not the case, the internal representation of ''f'' could be some append of append of... of append, and forcing would not be a constant time operation anymore.

* Circular buffer * Double-ended queue (deque) *
Priority queue In computer science, a priority queue is an abstract data-type similar to a regular queue or stack data structure in which each element additionally has a ''priority'' associated with it. In a priority queue, an element with high priority is s ...
* Queuing theory * Stack (abstract data type) – the "opposite" of a queue: LIFO (Last In First Out)

*