資料結構-Queue篇-1

發表於 2024-09-09 更新於 2025-01-14 分類於資料結構

前情提要

這篇文章是資料結構系列的第三篇，主要探討 Queue（佇列）的定義、抽象資料型別（ADT）、以及不同的實作方式。今天的內容包括線性隊列的問題與解決方法、環形隊列（Circular Queue）的兩種實現，以及鏈結串列實作的 Queue。文章還透過具體的範例程式碼展示了如何操作隊列中的新增（enqueue）和刪除（dequeue），幫助讀者更深入地理解隊列的運作與應用場景。

Queue 的定義

A queue is an ordered list in which all insertions take place at one end and all deletions take place at the opposite end. Given a queue \(Q = (a_0, a_1, \dotsb, a_{n-1})\), \(a_0\) 我們稱之為 front element, \(a_{n-1}\) 我們稱之為 rear element，那麼 Queue 有 \(\text{First-In-First-Out (FIFO)}\) 的性質。

Queue ADT

結構 : Queue
物件 : a finite ordered list with zero or more elements

函數定義

Queue CreateQ(max_queue_size) : 建立一個空的 queue 且最大容量為 max_queue_size。

Boolean IsFullQ(queue, max_queue_size) :

if (number of elements in queue == max_queue_size) 
    return TRUE
else 
    return FALSE

Boolean IsEmptyQ(queue) :

if (queue == CreateQ(max_queue_size)) 
    return TRUE
else 
    return FALSE

Queue 中新增 element

Queue AddQ(queue, item) 此為 Horowitz 的寫法 :

if (IsFullQ(queue)) 
    queue_full
else 
    insert item at rear of queue and return queue

ENQUEUE(Q, x) 此為 CLRS 中的寫法，此為 circular queue，稍後會說明到:

Q[Q.tail] = x
if Q.tail == Q.length
    Q.tail = 1
else 
    Q.tail = Q.tail + 1

Queue 中刪除 element

Element DeleteQ(queue) 此為 Horowitz 的寫法:

if (IsEmptyQ(queue)) 
    queue_empty
else 
    remove and return the item at front of queue

DEQUEUE(Q) 此為 CLRS 中的寫法 :

x = Q[Q.head]
if Q.head == Q.length
    Q.head = 1
else 
    Q.head = Q.head + 1
return x

Queue 的實作方式

Linear Array

實作程式碼

#include <iostream>
using namespace std;

class Queue {
private:
    int front, rear, size;
    int* arr;

public:
    Queue(int maxSize) {
        size = maxSize;
        arr = new int[size];
        front = -1;
        rear = -1;
    }

    ~Queue() {
        delete[] arr;
    }

    bool isFull() {
        return rear == size - 1;
    }

    bool isEmpty() {
        return front == rear;
    }

    void enqueue(int item) {
        if (isFull()) {
            cout << "Queue is full, shifting elements to the left..." << endl;
            shiftLeft();
        }
        if (rear == -1) {
            front = 0;
        }
        arr[++rear] = item;
        cout << "Enqueued: " << item << endl;
    }

    int dequeue() {
        if (isEmpty()) {
            cout << "Queue is empty." << endl;
            return -1;
        }
        int item = arr[front];
        front++;
        cout << "Dequeued: " << item << endl;
        return item;
    }

    void shiftLeft() {
        for (int i = front; i <= rear; i++) {
            arr[i - front] = arr[i];
        }
        rear = rear - front;
        front = 0;
    }
};

int main() {
    Queue q(3);

    q.enqueue(1);
    q.enqueue(2);
    q.enqueue(3);
    q.enqueue(4);
    q.dequeue();
    q.dequeue();
    q.enqueue(5);
    q.enqueue(6);

    return 0;
}

輸出內容

Enqueued: 1
Enqueued: 2
Enqueued: 3
Queue is full, shifting elements to the left...
Enqueued: 4
Dequeued: 1
Dequeued: 2
Enqueued: 5
Enqueued: 6

這個方法有一個嚴重的問題如下圖 (圖 2)，當 Job 進入或離開時，Queue 的索引（rear 和 front）會隨著作業的加入或刪除逐漸向右移動。當 \(rear\) 等於 \(\text{MAX_QUEUE_SIZE} - 1\) 時，這意味著隊列已滿。在這種情況下，必須通過 queue_full 操作將整個隊列左移，這樣 \(front\) 就會回到 \(-1\)，而 \(rear\) 也需要重新計算，以確保隊列正確排列。
這是一個典型的線性隊列，而不是循環隊列。當隊列滿了並且刪除了一些元素時，隊列的空間無法立即重用，因此需要整體左移來重新排列元素，這也是該隊列的限制之一。這樣花的時間是 \(O(\text{MAX_QUEUE_SIZE})\)，因為每次 queue_full 都要將所有元素往左移動。

Array Circular queue

法一

實作程式碼

#include <bits/stdc++.h>
using namespace std;

class CircularQueue {
private:
    int* queue;
    int size;
    int front, rear;

public:
    CircularQueue(int n) {
        size = n;
        queue = new int[size];
        front = 0;
        rear = 0;
    }

    bool isFull() {
        return (rear + 1) % size == front;
    }

    bool isEmpty() {
        return front == rear;
    }

    void enqueue(int value) {
        if (isFull()) {
            cout << "Queue is full" << endl;
            return;
        }
        queue[rear] = value;
        rear = (rear + 1) % size;
    }

    int dequeue() {
        if (isEmpty()) {
            cout << "Queue is empty" << endl;
            return -1;
        }
        int value = queue[front];
        front = (front + 1) % size;
        return value;
    }

    void display() {
        if (isEmpty()) {
            cout << "Queue is empty" << endl;
            return;
        }
        int i = front;
        while (i != rear) {
            cout << queue[i] << " ";
            i = (i + 1) % size;
        }
        cout << endl;
    }

    ~CircularQueue() {
        delete[] queue;
    }
};

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    CircularQueue q(5);
    q.enqueue(10);
    q.enqueue(20);
    q.enqueue(30);
    q.enqueue(40);

    q.display();

    q.dequeue();
    q.display();

    q.enqueue(50);
    q.display();

    q.enqueue(60);
}

輸出結果

10 20 30 40
20 30 40
20 30 40 50
Queue is full

使用這個方法最多可以利用 \(n-1\) 的空間。

法二

實作程式碼

#include <iostream>
using namespace std;

class CircularQueue {
private:
    int* queue;
    int size;
    int front;
    int rear;
    int tag;  

public:
    CircularQueue(int n) {
        size = n;
        queue = new int[size];
        front = 0;
        rear = 0;
        tag = 0;  
    }

    ~CircularQueue() {
        delete[] queue;
    }

    bool isFull() {
        return (front == rear && tag == 1);
    }

    bool isEmpty() {
        return (front == rear && tag == 0);
    }

    void enqueue(int value) {
        if (isFull()) {
            cout << "Queue is full" << endl;
            return;
        }
        queue[rear] = value;
        rear = (rear + 1) % size;
        tag = 1;  
    }

    int dequeue() {
        if (isEmpty()) {
            cout << "Queue is empty" << endl;
            return -1;  
        }
        int value = queue[front];
        front = (front + 1) % size;
        if (front == rear) {  
            tag = 0;
        }
        return value;
    }

    void display() {
    if (isEmpty()) {
        cout << "Queue is empty" << endl;
        return;
    }
    int i = front;
    cout << "Queue contents: ";
    while (i != rear || (i == rear && tag == 1)) {
        cout << queue[i] << " ";
        i = (i + 1) % size;
        if (i == front) break;  
    }
        cout << endl;
    }
};

int main() {
CircularQueue q(5);  

q.enqueue(10);
q.enqueue(20);
q.enqueue(30);
q.enqueue(40);
q.enqueue(50);
q.display();

cout << "Dequeued: " << q.dequeue() << endl;
q.display();

q.enqueue(60);
q.display();

return 0;
}

輸出內容

Queue contents: 10 20 30 40 50
Dequeued: 10
Queue contents: 20 30 40 50 10
Queue contents: 20 30 40 50 60

Link List queue

實作程式碼

#include <iostream>
using namespace std;

class Node {
public:
    int data;
    Node* next;

    Node(int val) {
    data = val;
    next = nullptr;
    }
};

class Queue {
private:
    Node* front;
    Node* rear;

public:
    Queue() {
        front = rear = nullptr;
    }

    bool isEmpty() {
        return front == nullptr;
    }

    void enqueue(int val) {
        Node* newNode = new Node(val);
        if (rear == nullptr) {
            front = rear = newNode;
            return;
        }
        rear->next = newNode;
        rear = newNode;
    }

    int dequeue() {
    if (isEmpty()) {
        cout << "Queue is empty" << endl;
        return -1;
    }
    Node* temp = front;
    front = front->next;

    if (front == nullptr) {
        rear = nullptr;
    }

    int dequeuedData = temp->data;
    delete temp;
    return dequeuedData;
    }

    int peek() {
    if (isEmpty()) {
        cout << "Queue is empty" << endl;
        return -1;
    }
    return front->data;
    }

    void display() {
    if (isEmpty()) {
        cout << "Queue is empty" << endl;
        return;
    }
    Node* current = front;
    while (current != nullptr) {
        cout << current->data << " -> ";
        current = current->next;
    }
    cout << "NULL" << endl;
    }
};

int main() {
    Queue q;
    q.enqueue(10);
    q.enqueue(20);
    q.enqueue(30);

    q.display();

    cout << "Dequeue: " << q.dequeue() << endl;
    q.display();
}