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# 5.3   Double-ended queue

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

![Operations in double-ended queue](deque.assets/deque_operations.png){ class="animation-figure" }

<p align="center"> Figure 5-7 &nbsp; Operations in double-ended queue </p>

## 5.3.1 &nbsp; 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.

<p align="center"> Table 5-3 &nbsp; Efficiency of double-ended queue operations </p>

<div class="center-table" markdown>

| 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    | $O(1)$          |

</div>

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

=== "Python"

    ```python title="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
    ```

=== "C++"

    ```cpp title="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();
    ```

=== "Java"

    ```java title="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();
    ```

=== "C#"

    ```csharp title="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;
    ```

=== "Go"

    ```go title="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
    ```

=== "Swift"

    ```swift title="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
    ```

=== "JS"

    ```javascript title="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;
    ```

=== "TS"

    ```typescript title="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;
    ```

=== "Dart"

    ```dart title="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;
    ```

=== "Rust"

    ```rust title="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();
    ```

=== "C"

    ```c title="deque.c"
    // C does not provide a built-in deque
    ```

=== "Kotlin"

    ```kotlin title="deque.kt"

    ```

=== "Zig"

    ```zig title="deque.zig"

    ```

??? pythontutor "Visualizing Code"

    https://pythontutor.com/render.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20deq.append%282%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E5%B0%BE%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E9%A6%96%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%20rear%20%3D%22,%20rear%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_rear%20%3D%22,%20pop_rear%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&cumulative=false&curInstr=3&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false

## 5.3.2 &nbsp; 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.

### 1. &nbsp; 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 Figure 5-8, 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.

=== "LinkedListDeque"
    ![Implementing Double-Ended Queue with Doubly Linked List for Enqueue and Dequeue Operations](deque.assets/linkedlist_deque_step1.png){ class="animation-figure" }

=== "pushLast()"
    ![linkedlist_deque_push_last](deque.assets/linkedlist_deque_step2_push_last.png){ class="animation-figure" }

=== "pushFirst()"
    ![linkedlist_deque_push_first](deque.assets/linkedlist_deque_step3_push_first.png){ class="animation-figure" }

=== "popLast()"
    ![linkedlist_deque_pop_last](deque.assets/linkedlist_deque_step4_pop_last.png){ class="animation-figure" }

=== "popFirst()"
    ![linkedlist_deque_pop_first](deque.assets/linkedlist_deque_step5_pop_first.png){ class="animation-figure" }

<p align="center"> Figure 5-8 &nbsp; Implementing Double-Ended Queue with Doubly Linked List for Enqueue and Dequeue Operations </p>

The implementation code is as follows:

=== "Python"

    ```python title="linkedlist_deque.py"
    class ListNode:
        """Double-linked list node"""

        def __init__(self, val: int):
            """Constructor"""
            self.val: int = val
            self.next: ListNode | None = None  # Reference to the next node
            self.prev: ListNode | None = None  # Reference to predecessor node

    class LinkedListDeque:
        """Double-ended queue class based on double-linked list"""

        def __init__(self):
            """Constructor"""
            self._front: ListNode | None = None  # Head node front
            self._rear: ListNode | None = None  # Tail node rear
            self._size: int = 0  # Length of the double-ended queue

        def size(self) -> int:
            """Get the length of the double-ended queue"""
            return self._size

        def is_empty(self) -> bool:
            """Determine if the double-ended queue is empty"""
            return self._size == 0

        def push(self, num: int, is_front: bool):
            """Enqueue operation"""
            node = ListNode(num)
            # If the list is empty, make front and rear both point to node
            if self.is_empty():
                self._front = self._rear = node
            # Front enqueue operation
            elif is_front:
                # Add node to the head of the list
                self._front.prev = node
                node.next = self._front
                self._front = node  # Update head node
            # Rear enqueue operation
            else:
                # Add node to the tail of the list
                self._rear.next = node
                node.prev = self._rear
                self._rear = node  # Update tail node
            self._size += 1  # Update queue length

        def push_first(self, num: int):
            """Front enqueue"""
            self.push(num, True)

        def push_last(self, num: int):
            """Rear enqueue"""
            self.push(num, False)

        def pop(self, is_front: bool) -> int:
            """Dequeue operation"""
            if self.is_empty():
                raise IndexError("Double-ended queue is empty")
            # Front dequeue operation
            if is_front:
                val: int = self._front.val  # Temporarily store the head node value
                # Remove head node
                fnext: ListNode | None = self._front.next
                if fnext != None:
                    fnext.prev = None
                    self._front.next = None
                self._front = fnext  # Update head node
            # Rear dequeue operation
            else:
                val: int = self._rear.val  # Temporarily store the tail node value
                # Remove tail node
                rprev: ListNode | None = self._rear.prev
                if rprev != None:
                    rprev.next = None
                    self._rear.prev = None
                self._rear = rprev  # Update tail node
            self._size -= 1  # Update queue length
            return val

        def pop_first(self) -> int:
            """Front dequeue"""
            return self.pop(True)

        def pop_last(self) -> int:
            """Rear dequeue"""
            return self.pop(False)

        def peek_first(self) -> int:
            """Access front element"""
            if self.is_empty():
                raise IndexError("Double-ended queue is empty")
            return self._front.val

        def peek_last(self) -> int:
            """Access rear element"""
            if self.is_empty():
                raise IndexError("Double-ended queue is empty")
            return self._rear.val

        def to_array(self) -> list[int]:
            """Return array for printing"""
            node = self._front
            res = [0] * self.size()
            for i in range(self.size()):
                res[i] = node.val
                node = node.next
            return res
    ```

=== "C++"

    ```cpp title="linkedlist_deque.cpp"
    [class]{DoublyListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Java"

    ```java title="linkedlist_deque.java"
    /* Double-linked list node */
    class ListNode {
        int val; // Node value
        ListNode next; // Reference to the next node
        ListNode prev; // Reference to predecessor node

        ListNode(int val) {
            this.val = val;
            prev = next = null;
        }
    }

    /* Double-ended queue class based on double-linked list */
    class LinkedListDeque {
        private ListNode front, rear; // Front node front, back node rear
        private int queSize = 0; // Length of the double-ended queue

        public LinkedListDeque() {
            front = rear = null;
        }

        /* Get the length of the double-ended queue */
        public int size() {
            return queSize;
        }

        /* Determine if the double-ended queue is empty */
        public boolean isEmpty() {
            return size() == 0;
        }

        /* Enqueue operation */
        private void push(int num, boolean isFront) {
            ListNode node = new ListNode(num);
            // If the list is empty, make front and rear both point to node
            if (isEmpty())
                front = rear = node;
            // Front enqueue operation
            else if (isFront) {
                // Add node to the head of the list
                front.prev = node;
                node.next = front;
                front = node; // Update head node
            // Rear enqueue operation
            } else {
                // Add node to the tail of the list
                rear.next = node;
                node.prev = rear;
                rear = node; // Update tail node
            }
            queSize++; // Update queue length
        }

        /* Front enqueue */
        public void pushFirst(int num) {
            push(num, true);
        }

        /* Rear enqueue */
        public void pushLast(int num) {
            push(num, false);
        }

        /* Dequeue operation */
        private int pop(boolean isFront) {
            if (isEmpty())
                throw new IndexOutOfBoundsException();
            int val;
            // Front dequeue operation
            if (isFront) {
                val = front.val; // Temporarily store the head node value
                // Remove head node
                ListNode fNext = front.next;
                if (fNext != null) {
                    fNext.prev = null;
                    front.next = null;
                }
                front = fNext; // Update head node
            // Rear dequeue operation
            } else {
                val = rear.val; // Temporarily store the tail node value
                // Remove tail node
                ListNode rPrev = rear.prev;
                if (rPrev != null) {
                    rPrev.next = null;
                    rear.prev = null;
                }
                rear = rPrev; // Update tail node
            }
            queSize--; // Update queue length
            return val;
        }

        /* Front dequeue */
        public int popFirst() {
            return pop(true);
        }

        /* Rear dequeue */
        public int popLast() {
            return pop(false);
        }

        /* Access front element */
        public int peekFirst() {
            if (isEmpty())
                throw new IndexOutOfBoundsException();
            return front.val;
        }

        /* Access rear element */
        public int peekLast() {
            if (isEmpty())
                throw new IndexOutOfBoundsException();
            return rear.val;
        }

        /* Return array for printing */
        public int[] toArray() {
            ListNode node = front;
            int[] res = new int[size()];
            for (int i = 0; i < res.length; i++) {
                res[i] = node.val;
                node = node.next;
            }
            return res;
        }
    }
    ```

=== "C#"

    ```csharp title="linkedlist_deque.cs"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Go"

    ```go title="linkedlist_deque.go"
    [class]{linkedListDeque}-[func]{}
    ```

=== "Swift"

    ```swift title="linkedlist_deque.swift"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "JS"

    ```javascript title="linkedlist_deque.js"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "TS"

    ```typescript title="linkedlist_deque.ts"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Dart"

    ```dart title="linkedlist_deque.dart"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Rust"

    ```rust title="linkedlist_deque.rs"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "C"

    ```c title="linkedlist_deque.c"
    [class]{DoublyListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Kotlin"

    ```kotlin title="linkedlist_deque.kt"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Ruby"

    ```ruby title="linkedlist_deque.rb"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

=== "Zig"

    ```zig title="linkedlist_deque.zig"
    [class]{ListNode}-[func]{}

    [class]{LinkedListDeque}-[func]{}
    ```

### 2. &nbsp; Implementation based on array

As shown in Figure 5-9, similar to implementing a queue with an array, we can also use a circular array to implement a double-ended queue.

=== "ArrayDeque"
    ![Implementing Double-Ended Queue with Array for Enqueue and Dequeue Operations](deque.assets/array_deque_step1.png){ class="animation-figure" }

=== "pushLast()"
    ![array_deque_push_last](deque.assets/array_deque_step2_push_last.png){ class="animation-figure" }

=== "pushFirst()"
    ![array_deque_push_first](deque.assets/array_deque_step3_push_first.png){ class="animation-figure" }

=== "popLast()"
    ![array_deque_pop_last](deque.assets/array_deque_step4_pop_last.png){ class="animation-figure" }

=== "popFirst()"
    ![array_deque_pop_first](deque.assets/array_deque_step5_pop_first.png){ class="animation-figure" }

<p align="center"> Figure 5-9 &nbsp; Implementing Double-Ended Queue with Array for Enqueue and Dequeue Operations </p>

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

=== "Python"

    ```python title="array_deque.py"
    class ArrayDeque:
        """Double-ended queue class based on circular array"""

        def __init__(self, capacity: int):
            """Constructor"""
            self._nums: list[int] = [0] * capacity
            self._front: int = 0
            self._size: int = 0

        def capacity(self) -> int:
            """Get the capacity of the double-ended queue"""
            return len(self._nums)

        def size(self) -> int:
            """Get the length of the double-ended queue"""
            return self._size

        def is_empty(self) -> bool:
            """Determine if the double-ended queue is empty"""
            return self._size == 0

        def index(self, i: int) -> int:
            """Calculate circular array index"""
            # Implement circular array by modulo operation
            # When i exceeds the tail of the array, return to the head
            # When i exceeds the head of the array, return to the tail
            return (i + self.capacity()) % self.capacity()

        def push_first(self, num: int):
            """Front enqueue"""
            if self._size == self.capacity():
                print("Double-ended queue is full")
                return
            # Move the front pointer one position to the left
            # Implement front crossing the head of the array to return to the tail by modulo operation
            self._front = self.index(self._front - 1)
            # Add num to the front
            self._nums[self._front] = num
            self._size += 1

        def push_last(self, num: int):
            """Rear enqueue"""
            if self._size == self.capacity():
                print("Double-ended queue is full")
                return
            # Calculate rear pointer, pointing to rear index + 1
            rear = self.index(self._front + self._size)
            # Add num to the rear
            self._nums[rear] = num
            self._size += 1

        def pop_first(self) -> int:
            """Front dequeue"""
            num = self.peek_first()
            # Move front pointer one position backward
            self._front = self.index(self._front + 1)
            self._size -= 1
            return num

        def pop_last(self) -> int:
            """Rear dequeue"""
            num = self.peek_last()
            self._size -= 1
            return num

        def peek_first(self) -> int:
            """Access front element"""
            if self.is_empty():
                raise IndexError("Double-ended queue is empty")
            return self._nums[self._front]

        def peek_last(self) -> int:
            """Access rear element"""
            if self.is_empty():
                raise IndexError("Double-ended queue is empty")
            # Calculate rear element index
            last = self.index(self._front + self._size - 1)
            return self._nums[last]

        def to_array(self) -> list[int]:
            """Return array for printing"""
            # Only convert elements within valid length range
            res = []
            for i in range(self._size):
                res.append(self._nums[self.index(self._front + i)])
            return res
    ```

=== "C++"

    ```cpp title="array_deque.cpp"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Java"

    ```java title="array_deque.java"
    /* Double-ended queue class based on circular array */
    class ArrayDeque {
        private int[] nums; // Array used to store elements of the double-ended queue
        private int front; // Front pointer, pointing to the front element
        private int queSize; // Length of the double-ended queue

        /* Constructor */
        public ArrayDeque(int capacity) {
            this.nums = new int[capacity];
            front = queSize = 0;
        }

        /* Get the capacity of the double-ended queue */
        public int capacity() {
            return nums.length;
        }

        /* Get the length of the double-ended queue */
        public int size() {
            return queSize;
        }

        /* Determine if the double-ended queue is empty */
        public boolean isEmpty() {
            return queSize == 0;
        }

        /* Calculate circular array index */
        private int index(int i) {
            // Implement circular array by modulo operation
            // When i exceeds the tail of the array, return to the head
            // When i exceeds the head of the array, return to the tail
            return (i + capacity()) % capacity();
        }

        /* Front enqueue */
        public void pushFirst(int num) {
            if (queSize == capacity()) {
                System.out.println("Double-ended queue is full");
                return;
            }
            // Move the front pointer one position to the left
            // Implement front crossing the head of the array to return to the tail by modulo operation
            front = index(front - 1);
            // Add num to the front
            nums[front] = num;
            queSize++;
        }

        /* Rear enqueue */
        public void pushLast(int num) {
            if (queSize == capacity()) {
                System.out.println("Double-ended queue is full");
                return;
            }
            // Calculate rear pointer, pointing to rear index + 1
            int rear = index(front + queSize);
            // Add num to the rear
            nums[rear] = num;
            queSize++;
        }

        /* Front dequeue */
        public int popFirst() {
            int num = peekFirst();
            // Move front pointer one position backward
            front = index(front + 1);
            queSize--;
            return num;
        }

        /* Rear dequeue */
        public int popLast() {
            int num = peekLast();
            queSize--;
            return num;
        }

        /* Access front element */
        public int peekFirst() {
            if (isEmpty())
                throw new IndexOutOfBoundsException();
            return nums[front];
        }

        /* Access rear element */
        public int peekLast() {
            if (isEmpty())
                throw new IndexOutOfBoundsException();
            // Calculate rear element index
            int last = index(front + queSize - 1);
            return nums[last];
        }

        /* Return array for printing */
        public int[] toArray() {
            // Only convert elements within valid length range
            int[] res = new int[queSize];
            for (int i = 0, j = front; i < queSize; i++, j++) {
                res[i] = nums[index(j)];
            }
            return res;
        }
    }
    ```

=== "C#"

    ```csharp title="array_deque.cs"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Go"

    ```go title="array_deque.go"
    [class]{arrayDeque}-[func]{}
    ```

=== "Swift"

    ```swift title="array_deque.swift"
    [class]{ArrayDeque}-[func]{}
    ```

=== "JS"

    ```javascript title="array_deque.js"
    [class]{ArrayDeque}-[func]{}
    ```

=== "TS"

    ```typescript title="array_deque.ts"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Dart"

    ```dart title="array_deque.dart"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Rust"

    ```rust title="array_deque.rs"
    [class]{ArrayDeque}-[func]{}
    ```

=== "C"

    ```c title="array_deque.c"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Kotlin"

    ```kotlin title="array_deque.kt"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Ruby"

    ```ruby title="array_deque.rb"
    [class]{ArrayDeque}-[func]{}
    ```

=== "Zig"

    ```zig title="array_deque.zig"
    [class]{ArrayDeque}-[func]{}
    ```

## 5.3.3 &nbsp; 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.