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Double-ended queue

In a queue, we can only delete elements from the head or add elements to the tail. As shown in the figure below, a double-ended queue (deque) offers more flexibility, allowing the addition or removal of elements at both the head and the tail.

Operations in double-ended queue

Common operations in double-ended queue

The common operations in a double-ended queue are listed below, and the names of specific methods depend on the programming language used.

Table   Efficiency of double-ended queue operations

Method Name Description Time Complexity
pushFirst() Add an element to the head \(O(1)\)
pushLast() Add an element to the tail \(O(1)\)
popFirst() Remove the first element \(O(1)\)
popLast() Remove the last element \(O(1)\)
peekFirst() Access the first element \(O(1)\)
peekLast() Access the last element \(O(1)\)

Similarly, we can directly use the double-ended queue classes implemented in programming languages:

deque.py
from collections import deque

# Initialize the deque
deq: deque[int] = deque()

# Enqueue elements
deq.append(2)      # Add to the tail
deq.append(5)
deq.append(4)
deq.appendleft(3)  # Add to the head
deq.appendleft(1)

# Access elements
front: int = deq[0]  # The first element
rear: int = deq[-1]  # The last element

# Dequeue elements
pop_front: int = deq.popleft()  # The first element dequeued
pop_rear: int = deq.pop()       # The last element dequeued

# Get the length of the deque
size: int = len(deq)

# Check if the deque is empty
is_empty: bool = len(deq) == 0
deque.cpp
/* Initialize the deque */
deque<int> deque;

/* Enqueue elements */
deque.push_back(2);   // Add to the tail
deque.push_back(5);
deque.push_back(4);
deque.push_front(3);  // Add to the head
deque.push_front(1);

/* Access elements */
int front = deque.front(); // The first element
int back = deque.back();   // The last element

/* Dequeue elements */
deque.pop_front();  // The first element dequeued
deque.pop_back();   // The last element dequeued

/* Get the length of the deque */
int size = deque.size();

/* Check if the deque is empty */
bool empty = deque.empty();
deque.java
/* Initialize the deque */
Deque<Integer> deque = new LinkedList<>();

/* Enqueue elements */
deque.offerLast(2);   // Add to the tail
deque.offerLast(5);
deque.offerLast(4);
deque.offerFirst(3);  // Add to the head
deque.offerFirst(1);

/* Access elements */
int peekFirst = deque.peekFirst();  // The first element
int peekLast = deque.peekLast();    // The last element

/* Dequeue elements */
int popFirst = deque.pollFirst();  // The first element dequeued
int popLast = deque.pollLast();    // The last element dequeued

/* Get the length of the deque */
int size = deque.size();

/* Check if the deque is empty */
boolean isEmpty = deque.isEmpty();
deque.cs
/* Initialize the deque */
// In C#, LinkedList is used as a deque
LinkedList<int> deque = new();

/* Enqueue elements */
deque.AddLast(2);   // Add to the tail
deque.AddLast(5);
deque.AddLast(4);
deque.AddFirst(3);  // Add to the head
deque.AddFirst(1);

/* Access elements */
int peekFirst = deque.First.Value;  // The first element
int peekLast = deque.Last.Value;    // The last element

/* Dequeue elements */
deque.RemoveFirst();  // The first element dequeued
deque.RemoveLast();   // The last element dequeued

/* Get the length of the deque */
int size = deque.Count;

/* Check if the deque is empty */
bool isEmpty = deque.Count == 0;
deque_test.go
/* Initialize the deque */
// In Go, use list as a deque
deque := list.New()

/* Enqueue elements */
deque.PushBack(2)      // Add to the tail
deque.PushBack(5)
deque.PushBack(4)
deque.PushFront(3)     // Add to the head
deque.PushFront(1)

/* Access elements */
front := deque.Front() // The first element
rear := deque.Back()   // The last element

/* Dequeue elements */
deque.Remove(front)    // The first element dequeued
deque.Remove(rear)     // The last element dequeued

/* Get the length of the deque */
size := deque.Len()

/* Check if the deque is empty */
isEmpty := deque.Len() == 0
deque.swift
/* Initialize the deque */
// Swift does not have a built-in deque class, so Array can be used as a deque
var deque: [Int] = []

/* Enqueue elements */
deque.append(2) // Add to the tail
deque.append(5)
deque.append(4)
deque.insert(3, at: 0) // Add to the head
deque.insert(1, at: 0)

/* Access elements */
let peekFirst = deque.first! // The first element
let peekLast = deque.last!   // The last element

/* Dequeue elements */
// Using Array, popFirst has a complexity of O(n)
let popFirst = deque.removeFirst() // The first element dequeued
let popLast = deque.removeLast()   // The last element dequeued

/* Get the length of the deque */
let size = deque.count

/* Check if the deque is empty */
let isEmpty = deque.isEmpty
deque.js
/* Initialize the deque */
// JavaScript does not have a built-in deque, so Array is used as a deque
const deque = [];

/* Enqueue elements */
deque.push(2);
deque.push(5);
deque.push(4);
// Note that unshift() has a time complexity of O(n) as it's an array
deque.unshift(3);
deque.unshift(1);

/* Access elements */
const peekFirst = deque[0]; // The first element
const peekLast = deque[deque.length - 1]; // The last element

/* Dequeue elements */
// Note that shift() has a time complexity of O(n) as it's an array
const popFront = deque.shift(); // The first element dequeued
const popBack = deque.pop();    // The last element dequeued

/* Get the length of the deque */
const size = deque.length;

/* Check if the deque is empty */
const isEmpty = size === 0;
deque.ts
/* Initialize the deque */
// TypeScript does not have a built-in deque, so Array is used as a deque
const deque: number[] = [];

/* Enqueue elements */
deque.push(2);
deque.push(5);
deque.push(4);
// Note that unshift() has a time complexity of O(n) as it's an array
deque.unshift(3);
deque.unshift(1);

/* Access elements */
const peekFirst: number = deque[0]; // The first element
const peekLast: number = deque[deque.length - 1]; // The last element

/* Dequeue elements */
// Note that shift() has a time complexity of O(n) as it's an array
const popFront: number = deque.shift() as number; // The first element dequeued
const popBack: number = deque.pop() as number;    // The last element dequeued

/* Get the length of the deque */
const size: number = deque.length;

/* Check if the deque is empty */
const isEmpty: boolean = size === 0;
deque.dart
/* Initialize the deque */
// In Dart, Queue is defined as a deque
Queue<int> deque = Queue<int>();

/* Enqueue elements */
deque.addLast(2);  // Add to the tail
deque.addLast(5);
deque.addLast(4);
deque.addFirst(3); // Add to the head
deque.addFirst(1);

/* Access elements */
int peekFirst = deque.first; // The first element
int peekLast = deque.last;   // The last element

/* Dequeue elements */
int popFirst = deque.removeFirst(); // The first element dequeued
int popLast = deque.removeLast();   // The last element dequeued

/* Get the length of the deque */
int size = deque.length;

/* Check if the deque is empty */
bool isEmpty = deque.isEmpty;
deque.rs
/* Initialize the deque */
let mut deque: VecDeque<u32> = VecDeque::new();

/* Enqueue elements */
deque.push_back(2);  // Add to the tail
deque.push_back(5);
deque.push_back(4);
deque.push_front(3); // Add to the head
deque.push_front(1);

/* Access elements */
if let Some(front) = deque.front() { // The first element
}
if let Some(rear) = deque.back() {   // The last element
}

/* Dequeue elements */
if let Some(pop_front) = deque.pop_front() { // The first element dequeued
}
if let Some(pop_rear) = deque.pop_back() {   // The last element dequeued
}

/* Get the length of the deque */
let size = deque.len();

/* Check if the deque is empty */
let is_empty = deque.is_empty();
deque.c
// C does not provide a built-in deque
deque.kt
deque.zig
Visualizing Code

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Implementing a double-ended queue *

The implementation of a double-ended queue is similar to that of a regular queue, it can be based on either a linked list or an array as the underlying data structure.

Implementation based on doubly linked list

Recall from the previous section that we used a regular singly linked list to implement a queue, as it conveniently allows for deleting from the head (corresponding to the dequeue operation) and adding new elements after the tail (corresponding to the enqueue operation).

For a double-ended queue, both the head and the tail can perform enqueue and dequeue operations. In other words, a double-ended queue needs to implement operations in the opposite direction as well. For this, we use a "doubly linked list" as the underlying data structure of the double-ended queue.

As shown in the figure below, we treat the head and tail nodes of the doubly linked list as the front and rear of the double-ended queue, respectively, and implement the functionality to add and remove nodes at both ends.

Implementing Double-Ended Queue with Doubly Linked List for Enqueue and Dequeue Operations

linkedlist_deque_push_last

linkedlist_deque_push_first

linkedlist_deque_pop_last

linkedlist_deque_pop_first

The implementation code is as follows:

[file]{linkedlist_deque}-[class]{linked_list_deque}-[func]{}

Implementation based on array

As shown in the figure below, similar to implementing a queue with an array, we can also use a circular array to implement a double-ended queue.

Implementing Double-Ended Queue with Array for Enqueue and Dequeue Operations

array_deque_push_last

array_deque_push_first

array_deque_pop_last

array_deque_pop_first

The implementation only needs to add methods for "front enqueue" and "rear dequeue":

[file]{array_deque}-[class]{array_deque}-[func]{}

Applications of double-ended queue

The double-ended queue combines the logic of both stacks and queues, thus, it can implement all their respective use cases while offering greater flexibility.

We know that software's "undo" feature is typically implemented using a stack: the system pushes each change operation onto the stack and then pops to implement undoing. However, considering the limitations of system resources, software often restricts the number of undo steps (for example, only allowing the last 50 steps). When the stack length exceeds 50, the software needs to perform a deletion operation at the bottom of the stack (the front of the queue). But a regular stack cannot perform this function, where a double-ended queue becomes necessary. Note that the core logic of "undo" still follows the Last-In-First-Out principle of a stack, but a double-ended queue can more flexibly implement some additional logic.