Double-ended queue

In computer science, a double-ended queue (abbreviated to deque,  ) is an abstract data type that generalizes a queue, for which elements can be added to or removed from either the front (head) or back (tail). It is also often called a head-tail linked list, though properly this refers to a specific data structure implementation of a deque (see below).

Naming conventions
Deque is sometimes written dequeue, but this use is generally deprecated in technical literature or technical writing because dequeue is also a verb meaning "to remove from a queue". Nevertheless, several libraries and some writers, such as Aho, Hopcroft, and Ullman in their textbook Data Structures and Algorithms, spell it dequeue. John Mitchell, author of Concepts in Programming Languages, also uses this terminology.

Distinctions and sub-types
This differs from the queue abstract data type or first in first out list (FIFO), where elements can only be added to one end and removed from the other. This general data class has some possible sub-types:


 * An input-restricted deque is one where deletion can be made from both ends, but insertion can be made at one end only.
 * An output-restricted deque is one where insertion can be made at both ends, but deletion can be made from one end only.

Both the basic and most common list types in computing, queues and stacks can be considered specializations of deques, and can be implemented using deques. A deque is a data structure that allows users to perform push and pop operations at both ends, providing flexibility in managing the order of elements.

Operations
The basic operations on a deque are enqueue and dequeue on either end. Also generally implemented are peek operations, which return the value at that end without dequeuing it.

Names vary between languages; major implementations include:

Implementations
There are at least two common ways to efficiently implement a deque: with a modified dynamic array or with a doubly linked list.

The dynamic array approach uses a variant of a dynamic array that can grow from both ends, sometimes called array deques. These array deques have all the properties of a dynamic array, such as constant-time random access, good locality of reference, and inefficient insertion/removal in the middle, with the addition of amortized constant-time insertion/removal at both ends, instead of just one end. Three common implementations include:
 * Storing deque contents in a circular buffer, and only resizing when the buffer becomes full. This decreases the frequency of resizings.
 * Allocating deque contents from the center of the underlying array, and resizing the underlying array when either end is reached. This approach may require more frequent resizings and waste more space, particularly when elements are only inserted at one end.
 * Storing contents in multiple smaller arrays, allocating additional arrays at the beginning or end as needed. Indexing is implemented by keeping a dynamic array containing pointers to each of the smaller arrays.

Purely functional implementation
Double-ended queues can also be implemented as a purely functional data structure. Two versions of the implementation exist. The first one, called 'real-time deque, is presented below. It allows the queue to be persistent with operations in $O(1)$ worst-case time, but requires lazy lists with memoization. The second one, with no lazy lists nor memoization is presented at the end of the sections. Its amortized time is $O(1)$ if the persistency is not used; but the worst-time complexity of an operation is $O(n)$ where $n$ is the number of elements in the double-ended queue.

Let us recall that, for a list,   denotes its length, that   represents an empty list and   represents the list whose head is   and whose tail is. The functions  and   return the list   without its first   elements, and the first   elements of , respectively. Or, if, they return the empty list and   respectively.

Real-time deques via lazy rebuilding and scheduling
A double-ended queue is represented as a sextuple  where   is a linked list which contains the front of the queue of length. Similarly,  is a linked list which represents the reverse of the rear of the queue, of length. Furthermore, it is assured that  and   - intuitively, it means that both the front and the rear contains between a third minus one and two thirds plus one of the elements. Finally,  and   are tails of   and of , they allow scheduling the moment where some lazy operations are forced. Note that, when a double-ended queue contains  elements in the front list and   elements in the rear list, then the inequality invariant remains satisfied after   insertions and   deletions when. That is, at most  operations can happen between each rebalancing.

Let us first give an implementation of the various operations that affect the front of the deque - cons, head and tail. Those implementations do not necessarily respect the invariant. In a second time we'll explain how to modify a deque which does not satisfy the invariant into one which satisfies it. However, they use the invariant, in that if the front is empty then the rear has at most one element. The operations affecting the rear of the list are defined similarly by symmetry.

It remains to explain how to define a method  that rebalance the deque if   or   broke the invariant. The method  and   can be defined by first applying   and   and then applying.

where  return the concatenation of   and of. That is put into   the content of   and the  content of   that is not already in. Since dropping  elements takes $$O(n)$$ time, we use laziness to ensure that elements are dropped two by two, with two drops being done during each   and each   operation.

where  is a function that returns the front, followed by the middle reversed, followed by the rear. This function is also defined using laziness to ensure that it can be computed step by step, with one step executed during each  and   and taking a constant time. This function uses the invariant that  is 2 or 3.

where  is the function concatenating two lists.

Implementation without laziness
Note that, without the lazy part of the implementation, this would be a non-persistent implementation of queue in $O(1)$ amortized time. In this case, the lists  and   could be removed from the representation of the double-ended queue.

Language support
Ada's containers provides the generic packages  and , for the dynamic array and linked list implementations, respectively.

C++'s Standard Template Library provides the class templates  and , for the multiple array and linked list implementations, respectively.

As of Java 6, Java's Collections Framework provides a new interface that provides the functionality of insertion and removal at both ends. It is implemented by classes such as (also new in Java 6) and, providing the dynamic array and linked list implementations, respectively. However, the, contrary to its name, does not support random access.

Javascript's Array prototype & Perl's arrays have native support for both removing (shift and pop) and adding (unshift and push) elements on both ends.

Python 2.4 introduced the  module with support for deque objects. It is implemented using a doubly linked list of fixed-length subarrays.

As of PHP 5.3, PHP's SPL extension contains the 'SplDoublyLinkedList' class that can be used to implement Deque datastructures. Previously to make a Deque structure the array functions array_shift/unshift/pop/push had to be used instead.

GHC's Data.Sequence module implements an efficient, functional deque structure in Haskell. The implementation uses 2–3 finger trees annotated with sizes. There are other (fast) possibilities to implement purely functional (thus also persistent) double queues (most using heavily lazy evaluation). Kaplan and Tarjan were the first to implement optimal confluently persistent catenable deques. Their implementation was strictly purely functional in the sense that it did not use lazy evaluation. Okasaki simplified the data structure by using lazy evaluation with a bootstrapped data structure and degrading the performance bounds from worst-case to amortized. Kaplan, Okasaki, and Tarjan produced a simpler, non-bootstrapped, amortized version that can be implemented either using lazy evaluation or more efficiently using mutation in a broader but still restricted fashion. Mihaescu and Tarjan created a simpler (but still highly complex) strictly purely functional implementation of catenable deques, and also a much simpler implementation of strictly purely functional non-catenable deques, both of which have optimal worst-case bounds.

Rust's  includes VecDeque which implements a double-ended queue using a growable ring buffer.

Complexity

 * In a doubly-linked list implementation and assuming no allocation/deallocation overhead, the time complexity of all deque operations is O(1). Additionally, the time complexity of insertion or deletion in the middle, given an iterator, is O(1); however, the time complexity of random access by index is O(n).
 * In a growing array, the amortized time complexity of all deque operations is O(1). Additionally, the time complexity of random access by index is O(1); but the time complexity of insertion or deletion in the middle is O(n).

Applications
One example where a deque can be used is the work stealing algorithm. This algorithm implements task scheduling for several processors. A separate deque with threads to be executed is maintained for each processor. To execute the next thread, the processor gets the first element from the deque (using the "remove first element" deque operation). If the current thread forks, it is put back to the front of the deque ("insert element at front") and a new thread is executed. When one of the processors finishes execution of its own threads (i.e. its deque is empty), it can "steal" a thread from another processor: it gets the last element from the deque of another processor ("remove last element") and executes it. The work stealing algorithm is used by Intel's Threading Building Blocks (TBB) library for parallel programming.