5.2 Queue¶
A queue is a linear data structure that follows the First-In-First-Out (FIFO) rule. As the name suggests, a queue simulates the phenomenon of lining up, where newcomers join the queue at the rear, and the person at the front leaves the queue first.
As shown in Figure 5-4, we call the front of the queue the "head" and the back the "tail." The operation of adding elements to the rear of the queue is termed "enqueue," and the operation of removing elements from the front is termed "dequeue."
Figure 5-4 Queue's first-in-first-out rule
5.2.1 Common operations on queue¶
The common operations on a queue are shown in Table 5-2. Note that method names may vary across different programming languages. Here, we use the same naming convention as that used for stacks.
Table 5-2 Efficiency of queue operations
| Method Name | Description | Time Complexity |
|---|---|---|
push() |
Enqueue an element, add it to the tail | \(O(1)\) |
pop() |
Dequeue the head element | \(O(1)\) |
peek() |
Access the head element | \(O(1)\) |
We can directly use the ready-made queue classes in programming languages:
queue.py
from collections import deque
# Initialize the queue
# In Python, we generally use the deque class as a queue
# Although queue.Queue() is a pure queue class, it's not very user-friendly, so it's not recommended
que: deque[int] = deque()
# Enqueue elements
que.append(1)
que.append(3)
que.append(2)
que.append(5)
que.append(4)
# Access the first element
front: int = que[0]
# Dequeue an element
pop: int = que.popleft()
# Get the length of the queue
size: int = len(que)
# Check if the queue is empty
is_empty: bool = len(que) == 0
queue.cpp
/* Initialize the queue */
queue<int> queue;
/* Enqueue elements */
queue.push(1);
queue.push(3);
queue.push(2);
queue.push(5);
queue.push(4);
/* Access the first element*/
int front = queue.front();
/* Dequeue an element */
queue.pop();
/* Get the length of the queue */
int size = queue.size();
/* Check if the queue is empty */
bool empty = queue.empty();
queue.java
/* Initialize the queue */
Queue<Integer> queue = new LinkedList<>();
/* Enqueue elements */
queue.offer(1);
queue.offer(3);
queue.offer(2);
queue.offer(5);
queue.offer(4);
/* Access the first element */
int peek = queue.peek();
/* Dequeue an element */
int pop = queue.poll();
/* Get the length of the queue */
int size = queue.size();
/* Check if the queue is empty */
boolean isEmpty = queue.isEmpty();
queue.cs
/* Initialize the queue */
Queue<int> queue = new();
/* Enqueue elements */
queue.Enqueue(1);
queue.Enqueue(3);
queue.Enqueue(2);
queue.Enqueue(5);
queue.Enqueue(4);
/* Access the first element */
int peek = queue.Peek();
/* Dequeue an element */
int pop = queue.Dequeue();
/* Get the length of the queue */
int size = queue.Count;
/* Check if the queue is empty */
bool isEmpty = queue.Count == 0;
queue_test.go
/* Initialize the queue */
// In Go, use list as a queue
queue := list.New()
/* Enqueue elements */
queue.PushBack(1)
queue.PushBack(3)
queue.PushBack(2)
queue.PushBack(5)
queue.PushBack(4)
/* Access the first element */
peek := queue.Front()
/* Dequeue an element */
pop := queue.Front()
queue.Remove(pop)
/* Get the length of the queue */
size := queue.Len()
/* Check if the queue is empty */
isEmpty := queue.Len() == 0
queue.swift
/* Initialize the queue */
// Swift does not have a built-in queue class, so Array can be used as a queue
var queue: [Int] = []
/* Enqueue elements */
queue.append(1)
queue.append(3)
queue.append(2)
queue.append(5)
queue.append(4)
/* Access the first element */
let peek = queue.first!
/* Dequeue an element */
// Since it's an array, removeFirst has a complexity of O(n)
let pool = queue.removeFirst()
/* Get the length of the queue */
let size = queue.count
/* Check if the queue is empty */
let isEmpty = queue.isEmpty
queue.js
/* Initialize the queue */
// JavaScript does not have a built-in queue, so Array can be used as a queue
const queue = [];
/* Enqueue elements */
queue.push(1);
queue.push(3);
queue.push(2);
queue.push(5);
queue.push(4);
/* Access the first element */
const peek = queue[0];
/* Dequeue an element */
// Since the underlying structure is an array, shift() method has a time complexity of O(n)
const pop = queue.shift();
/* Get the length of the queue */
const size = queue.length;
/* Check if the queue is empty */
const empty = queue.length === 0;
queue.ts
/* Initialize the queue */
// TypeScript does not have a built-in queue, so Array can be used as a queue
const queue: number[] = [];
/* Enqueue elements */
queue.push(1);
queue.push(3);
queue.push(2);
queue.push(5);
queue.push(4);
/* Access the first element */
const peek = queue[0];
/* Dequeue an element */
// Since the underlying structure is an array, shift() method has a time complexity of O(n)
const pop = queue.shift();
/* Get the length of the queue */
const size = queue.length;
/* Check if the queue is empty */
const empty = queue.length === 0;
queue.dart
/* Initialize the queue */
// In Dart, the Queue class is a double-ended queue but can be used as a queue
Queue<int> queue = Queue();
/* Enqueue elements */
queue.add(1);
queue.add(3);
queue.add(2);
queue.add(5);
queue.add(4);
/* Access the first element */
int peek = queue.first;
/* Dequeue an element */
int pop = queue.removeFirst();
/* Get the length of the queue */
int size = queue.length;
/* Check if the queue is empty */
bool isEmpty = queue.isEmpty;
queue.rs
/* Initialize the double-ended queue */
// In Rust, use a double-ended queue as a regular queue
let mut deque: VecDeque<u32> = VecDeque::new();
/* Enqueue elements */
deque.push_back(1);
deque.push_back(3);
deque.push_back(2);
deque.push_back(5);
deque.push_back(4);
/* Access the first element */
if let Some(front) = deque.front() {
}
/* Dequeue an element */
if let Some(pop) = deque.pop_front() {
}
/* Get the length of the queue */
let size = deque.len();
/* Check if the queue is empty */
let is_empty = deque.is_empty();
Code Visualization
5.2.2 Implementing a queue¶
To implement a queue, we need a data structure that allows adding elements at one end and removing them at the other. Both linked lists and arrays meet this requirement.
1. Implementation based on a linked list¶
As shown in Figure 5-5, we can consider the "head node" and "tail node" of a linked list as the "front" and "rear" of the queue, respectively. It is stipulated that nodes can only be added at the rear and removed at the front.
Figure 5-5 Implementing Queue with Linked List for Enqueue and Dequeue Operations
Below is the code for implementing a queue using a linked list:
linkedlist_queue.py
class LinkedListQueue:
"""Queue class based on 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
def size(self) -> int:
"""Get the length of the queue"""
return self._size
def is_empty(self) -> bool:
"""Determine if the queue is empty"""
return self._size == 0
def push(self, num: int):
"""Enqueue"""
# Add num behind the tail node
node = ListNode(num)
# If the queue is empty, make the head and tail nodes both point to that node
if self._front is None:
self._front = node
self._rear = node
# If the queue is not empty, add that node behind the tail node
else:
self._rear.next = node
self._rear = node
self._size += 1
def pop(self) -> int:
"""Dequeue"""
num = self.peek()
# Remove head node
self._front = self._front.next
self._size -= 1
return num
def peek(self) -> int:
"""Access front element"""
if self.is_empty():
raise IndexError("Queue is empty")
return self._front.val
def to_list(self) -> list[int]:
"""Convert to a list for printing"""
queue = []
temp = self._front
while temp:
queue.append(temp.val)
temp = temp.next
return queue
linkedlist_queue.cpp
/* Queue class based on linked list */
class LinkedListQueue {
private:
ListNode *front, *rear; // Front node front, back node rear
int queSize;
public:
LinkedListQueue() {
front = nullptr;
rear = nullptr;
queSize = 0;
}
~LinkedListQueue() {
// Traverse the linked list, remove nodes, free memory
freeMemoryLinkedList(front);
}
/* Get the length of the queue */
int size() {
return queSize;
}
/* Determine if the queue is empty */
bool isEmpty() {
return queSize == 0;
}
/* Enqueue */
void push(int num) {
// Add num behind the tail node
ListNode *node = new ListNode(num);
// If the queue is empty, make the head and tail nodes both point to that node
if (front == nullptr) {
front = node;
rear = node;
}
// If the queue is not empty, add that node behind the tail node
else {
rear->next = node;
rear = node;
}
queSize++;
}
/* Dequeue */
int pop() {
int num = peek();
// Remove head node
ListNode *tmp = front;
front = front->next;
// Free memory
delete tmp;
queSize--;
return num;
}
/* Access front element */
int peek() {
if (size() == 0)
throw out_of_range("Queue is empty");
return front->val;
}
/* Convert the linked list to Vector and return */
vector<int> toVector() {
ListNode *node = front;
vector<int> res(size());
for (int i = 0; i < res.size(); i++) {
res[i] = node->val;
node = node->next;
}
return res;
}
};
linkedlist_queue.java
/* Queue class based on linked list */
class LinkedListQueue {
private ListNode front, rear; // Front node front, back node rear
private int queSize = 0;
public LinkedListQueue() {
front = null;
rear = null;
}
/* Get the length of the queue */
public int size() {
return queSize;
}
/* Determine if the queue is empty */
public boolean isEmpty() {
return size() == 0;
}
/* Enqueue */
public void push(int num) {
// Add num behind the tail node
ListNode node = new ListNode(num);
// If the queue is empty, make the head and tail nodes both point to that node
if (front == null) {
front = node;
rear = node;
// If the queue is not empty, add that node behind the tail node
} else {
rear.next = node;
rear = node;
}
queSize++;
}
/* Dequeue */
public int pop() {
int num = peek();
// Remove head node
front = front.next;
queSize--;
return num;
}
/* Access front element */
public int peek() {
if (isEmpty())
throw new IndexOutOfBoundsException();
return front.val;
}
/* Convert the linked list to Array and return */
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;
}
}
2. Implementation based on an array¶
Deleting the first element in an array has a time complexity of \(O(n)\), which would make the dequeue operation inefficient. However, this problem can be cleverly avoided as follows.
We use a variable front to indicate the index of the front element and maintain a variable size to record the queue's length. Define rear = front + size, which points to the position immediately following the tail element.
With this design, the effective interval of elements in the array is [front, rear - 1]. The implementation methods for various operations are shown in Figure 5-6.
- Enqueue operation: Assign the input element to the
rearindex and increasesizeby 1. - Dequeue operation: Simply increase
frontby 1 and decreasesizeby 1.
Both enqueue and dequeue operations only require a single operation, each with a time complexity of \(O(1)\).
Figure 5-6 Implementing Queue with Array for Enqueue and Dequeue Operations
You might notice a problem: as enqueue and dequeue operations are continuously performed, both front and rear move to the right and will eventually reach the end of the array and can't move further. To resolve this, we can treat the array as a "circular array" where connecting the end of the array back to its beginning.
In a circular array, front or rear needs to loop back to the start of the array upon reaching the end. This cyclical pattern can be achieved with a "modulo operation" as shown in the code below:
array_queue.py
class ArrayQueue:
"""Queue class based on circular array"""
def __init__(self, size: int):
"""Constructor"""
self._nums: list[int] = [0] * size # Array for storing queue elements
self._front: int = 0 # Front pointer, pointing to the front element
self._size: int = 0 # Queue length
def capacity(self) -> int:
"""Get the capacity of the queue"""
return len(self._nums)
def size(self) -> int:
"""Get the length of the queue"""
return self._size
def is_empty(self) -> bool:
"""Determine if the queue is empty"""
return self._size == 0
def push(self, num: int):
"""Enqueue"""
if self._size == self.capacity():
raise IndexError("Queue is full")
# Calculate rear pointer, pointing to rear index + 1
# Use modulo operation to wrap the rear pointer from the end of the array back to the start
rear: int = (self._front + self._size) % self.capacity()
# Add num to the rear
self._nums[rear] = num
self._size += 1
def pop(self) -> int:
"""Dequeue"""
num: int = self.peek()
# Move front pointer one position backward, returning to the head of the array if it exceeds the tail
self._front = (self._front + 1) % self.capacity()
self._size -= 1
return num
def peek(self) -> int:
"""Access front element"""
if self.is_empty():
raise IndexError("Queue is empty")
return self._nums[self._front]
def to_list(self) -> list[int]:
"""Return array for printing"""
res = [0] * self.size()
j: int = self._front
for i in range(self.size()):
res[i] = self._nums[(j % self.capacity())]
j += 1
return res
array_queue.cpp
/* Queue class based on circular array */
class ArrayQueue {
private:
int *nums; // Array for storing queue elements
int front; // Front pointer, pointing to the front element
int queSize; // Queue length
int queCapacity; // Queue capacity
public:
ArrayQueue(int capacity) {
// Initialize an array
nums = new int[capacity];
queCapacity = capacity;
front = queSize = 0;
}
~ArrayQueue() {
delete[] nums;
}
/* Get the capacity of the queue */
int capacity() {
return queCapacity;
}
/* Get the length of the queue */
int size() {
return queSize;
}
/* Determine if the queue is empty */
bool isEmpty() {
return size() == 0;
}
/* Enqueue */
void push(int num) {
if (queSize == queCapacity) {
cout << "Queue is full" << endl;
return;
}
// Calculate rear pointer, pointing to rear index + 1
// Use modulo operation to wrap the rear pointer from the end of the array back to the start
int rear = (front + queSize) % queCapacity;
// Add num to the rear
nums[rear] = num;
queSize++;
}
/* Dequeue */
int pop() {
int num = peek();
// Move front pointer one position backward, returning to the head of the array if it exceeds the tail
front = (front + 1) % queCapacity;
queSize--;
return num;
}
/* Access front element */
int peek() {
if (isEmpty())
throw out_of_range("Queue is empty");
return nums[front];
}
/* Convert array to Vector and return */
vector<int> toVector() {
// Only convert elements within valid length range
vector<int> arr(queSize);
for (int i = 0, j = front; i < queSize; i++, j++) {
arr[i] = nums[j % queCapacity];
}
return arr;
}
};
array_queue.java
/* Queue class based on circular array */
class ArrayQueue {
private int[] nums; // Array for storing queue elements
private int front; // Front pointer, pointing to the front element
private int queSize; // Queue length
public ArrayQueue(int capacity) {
nums = new int[capacity];
front = queSize = 0;
}
/* Get the capacity of the queue */
public int capacity() {
return nums.length;
}
/* Get the length of the queue */
public int size() {
return queSize;
}
/* Determine if the queue is empty */
public boolean isEmpty() {
return queSize == 0;
}
/* Enqueue */
public void push(int num) {
if (queSize == capacity()) {
System.out.println("Queue is full");
return;
}
// Calculate rear pointer, pointing to rear index + 1
// Use modulo operation to wrap the rear pointer from the end of the array back to the start
int rear = (front + queSize) % capacity();
// Add num to the rear
nums[rear] = num;
queSize++;
}
/* Dequeue */
public int pop() {
int num = peek();
// Move front pointer one position backward, returning to the head of the array if it exceeds the tail
front = (front + 1) % capacity();
queSize--;
return num;
}
/* Access front element */
public int peek() {
if (isEmpty())
throw new IndexOutOfBoundsException();
return nums[front];
}
/* Return array */
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[j % capacity()];
}
return res;
}
}
The above implementation of the queue still has its limitations: its length is fixed. However, this issue is not difficult to resolve. We can replace the array with a dynamic array that can expand itself if needed. Interested readers can try to implement this themselves.
The comparison of the two implementations is consistent with that of the stack and is not repeated here.
5.2.3 Typical applications of queue¶
- Amazon orders: After shoppers place orders, these orders join a queue, and the system processes them in order. During events like Singles' Day, a massive number of orders are generated in a short time, making high concurrency a key challenge for engineers.
- Various to-do lists: Any scenario requiring a "first-come, first-served" functionality, such as a printer's task queue or a restaurant's food delivery queue, can effectively maintain the order of processing with a queue.