2023-02-28 20:03:53 +08:00
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# 队列
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2022-11-22 17:47:26 +08:00
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「队列 Queue」是一种遵循「先入先出 first in, first out」数据操作规则的线性数据结构。顾名思义,队列模拟的是排队现象,即外面的人不断加入队列尾部,而处于队列头部的人不断地离开。
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我们将队列头部称为「队首」,队列尾部称为「队尾」,将把元素加入队尾的操作称为「入队」,删除队首元素的操作称为「出队」。
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2023-02-26 18:18:34 +08:00
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![队列的先入先出规则](queue.assets/queue_operations.png)
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2022-11-22 17:47:26 +08:00
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2023-02-16 03:39:01 +08:00
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## 队列常用操作
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2022-11-22 17:47:26 +08:00
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2023-03-13 21:58:21 +08:00
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队列的常用操作见下表。需要注意,不同编程语言的方法名是不同的,在这里我们采用与栈相同的方法命名。
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2022-11-22 17:47:26 +08:00
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<div class="center-table" markdown>
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2023-02-02 01:43:01 +08:00
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| 方法名 | 描述 | 时间复杂度 |
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| --------- | -------------------------- | -------- |
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| push() | 元素入队,即将元素添加至队尾 | $O(1)$ |
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2023-03-13 21:58:21 +08:00
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| pop() | 队首元素出队 | $O(1)$ |
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| peek() | 访问队首元素 | $O(1)$ |
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2022-11-22 17:47:26 +08:00
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</div>
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我们可以直接使用编程语言实现好的队列类。
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=== "Java"
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```java title="queue.java"
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/* 初始化队列 */
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Queue<Integer> queue = new LinkedList<>();
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2023-01-07 20:34:32 +08:00
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2022-11-22 17:47:26 +08:00
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/* 元素入队 */
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queue.offer(1);
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queue.offer(3);
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queue.offer(2);
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queue.offer(5);
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queue.offer(4);
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2023-01-07 20:34:32 +08:00
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2022-11-22 17:47:26 +08:00
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/* 访问队首元素 */
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int peek = queue.peek();
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2023-01-07 20:34:32 +08:00
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2022-11-22 17:47:26 +08:00
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/* 元素出队 */
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2023-03-13 21:58:21 +08:00
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int pop = queue.poll();
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2023-01-07 20:34:32 +08:00
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2022-11-22 17:47:26 +08:00
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/* 获取队列的长度 */
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int size = queue.size();
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2023-01-07 20:34:32 +08:00
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2022-11-22 17:47:26 +08:00
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/* 判断队列是否为空 */
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boolean isEmpty = queue.isEmpty();
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```
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=== "C++"
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```cpp title="queue.cpp"
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2022-11-30 03:46:53 +08:00
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/* 初始化队列 */
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queue<int> queue;
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/* 元素入队 */
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queue.push(1);
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queue.push(3);
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queue.push(2);
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queue.push(5);
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queue.push(4);
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/* 访问队首元素 */
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int front = queue.front();
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2022-11-22 17:47:26 +08:00
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2022-11-30 03:46:53 +08:00
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/* 元素出队 */
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queue.pop();
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/* 获取队列的长度 */
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int size = queue.size();
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/* 判断队列是否为空 */
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bool empty = queue.empty();
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2022-11-22 17:47:26 +08:00
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```
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=== "Python"
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```python title="queue.py"
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2022-12-02 00:09:34 +08:00
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""" 初始化队列 """
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2022-12-19 21:38:31 +08:00
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# 在 Python 中,我们一般将双向队列类 deque 看作队列使用
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2022-12-02 00:09:34 +08:00
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# 虽然 queue.Queue() 是纯正的队列类,但不太好用,因此不建议
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2023-03-12 18:49:52 +08:00
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que: Deque[int] = collections.deque()
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2023-01-07 20:34:32 +08:00
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2022-12-02 00:09:34 +08:00
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""" 元素入队 """
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que.append(1)
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que.append(3)
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que.append(2)
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que.append(5)
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que.append(4)
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2023-01-07 20:34:32 +08:00
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2022-12-02 00:09:34 +08:00
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""" 访问队首元素 """
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2023-03-12 18:49:52 +08:00
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front: int = que[0];
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2023-01-07 20:34:32 +08:00
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2022-12-02 00:09:34 +08:00
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""" 元素出队 """
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2023-03-12 18:49:52 +08:00
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pop: int = que.popleft()
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2023-01-07 20:34:32 +08:00
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2022-12-02 00:09:34 +08:00
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""" 获取队列的长度 """
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2023-03-12 18:49:52 +08:00
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size: int = len(que)
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2023-01-07 20:34:32 +08:00
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2022-12-02 00:09:34 +08:00
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""" 判断队列是否为空 """
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2023-03-12 18:49:52 +08:00
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is_empty: bool = len(que) == 0
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2022-11-22 17:47:26 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "Go"
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2022-12-15 23:30:53 +08:00
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```go title="queue_test.go"
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2022-12-03 20:25:24 +08:00
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/* 初始化队列 */
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// 在 Go 中,将 list 作为队列来使用
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queue := list.New()
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2023-01-07 20:34:32 +08:00
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2022-12-03 20:25:24 +08:00
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/* 元素入队 */
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queue.PushBack(1)
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queue.PushBack(3)
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queue.PushBack(2)
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queue.PushBack(5)
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queue.PushBack(4)
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2023-01-07 20:34:32 +08:00
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2022-12-03 20:25:24 +08:00
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/* 访问队首元素 */
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peek := queue.Front()
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2023-01-07 20:34:32 +08:00
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2022-12-03 20:25:24 +08:00
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/* 元素出队 */
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2023-03-13 21:58:21 +08:00
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pop := queue.Front()
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queue.Remove(pop)
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2023-01-07 20:34:32 +08:00
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2022-12-03 20:25:24 +08:00
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/* 获取队列的长度 */
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size := queue.Len()
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2023-01-07 20:34:32 +08:00
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2022-12-03 20:25:24 +08:00
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/* 判断队列是否为空 */
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isEmpty := queue.Len() == 0
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2022-12-03 01:31:29 +08:00
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```
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=== "JavaScript"
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2023-02-08 04:27:55 +08:00
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```javascript title="queue.js"
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2022-12-05 22:07:41 +08:00
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/* 初始化队列 */
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2022-12-20 21:33:14 +08:00
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// JavaScript 没有内置的队列,可以把 Array 当作队列来使用
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2022-12-05 22:06:31 +08:00
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const queue = [];
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 元素入队 */
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queue.push(1);
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queue.push(3);
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queue.push(2);
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queue.push(5);
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queue.push(4);
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 访问队首元素 */
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const peek = queue[0];
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 元素出队 */
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2022-12-20 21:33:14 +08:00
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// 底层是数组,因此 shift() 方法的时间复杂度为 O(n)
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2023-03-13 21:58:21 +08:00
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const pop = queue.shift();
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 获取队列的长度 */
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const size = queue.length;
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 判断队列是否为空 */
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const empty = queue.length === 0;
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2022-12-03 01:31:29 +08:00
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```
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=== "TypeScript"
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```typescript title="queue.ts"
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2022-12-05 22:06:31 +08:00
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/* 初始化队列 */
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2022-12-05 22:56:38 +08:00
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// TypeScript 没有内置的队列,可以把 Array 当作队列来使用
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2022-12-05 22:06:31 +08:00
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const queue: number[] = [];
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 元素入队 */
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queue.push(1);
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queue.push(3);
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queue.push(2);
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queue.push(5);
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queue.push(4);
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 访问队首元素 */
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const peek = queue[0];
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 元素出队 */
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2022-12-20 21:33:14 +08:00
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// 底层是数组,因此 shift() 方法的时间复杂度为 O(n)
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2023-03-13 21:58:21 +08:00
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const pop = queue.shift();
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 获取队列的长度 */
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const size = queue.length;
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2023-01-07 20:34:32 +08:00
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2022-12-05 22:06:31 +08:00
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/* 判断队列是否为空 */
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const empty = queue.length === 0;
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2022-12-03 01:31:29 +08:00
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```
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=== "C"
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```c title="queue.c"
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2023-01-07 20:34:32 +08:00
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2022-12-03 01:31:29 +08:00
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```
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=== "C#"
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```csharp title="queue.cs"
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2022-12-23 15:42:02 +08:00
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/* 初始化队列 */
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Queue<int> queue = new();
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2023-01-07 20:34:32 +08:00
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2022-12-23 15:42:02 +08:00
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/* 元素入队 */
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queue.Enqueue(1);
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queue.Enqueue(3);
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queue.Enqueue(2);
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queue.Enqueue(5);
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queue.Enqueue(4);
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2023-01-07 20:34:32 +08:00
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2022-12-23 15:42:02 +08:00
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/* 访问队首元素 */
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int peek = queue.Peek();
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2023-01-07 20:34:32 +08:00
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2022-12-23 15:42:02 +08:00
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/* 元素出队 */
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2023-03-13 21:58:21 +08:00
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int pop = queue.Dequeue();
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2023-01-07 20:34:32 +08:00
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2022-12-23 15:42:02 +08:00
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/* 获取队列的长度 */
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int size = queue.Count();
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2023-01-07 20:34:32 +08:00
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2022-12-23 15:42:02 +08:00
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/* 判断队列是否为空 */
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bool isEmpty = queue.Count() == 0;
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2022-12-03 01:31:29 +08:00
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```
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2023-01-08 19:41:05 +08:00
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=== "Swift"
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```swift title="queue.swift"
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2023-01-11 23:48:10 +08:00
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/* 初始化队列 */
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// Swift 没有内置的队列类,可以把 Array 当作队列来使用
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var queue: [Int] = []
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2023-01-19 02:04:21 +08:00
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2023-01-11 23:48:10 +08:00
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/* 元素入队 */
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queue.append(1)
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queue.append(3)
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queue.append(2)
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queue.append(5)
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queue.append(4)
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2023-01-19 02:04:21 +08:00
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2023-01-11 23:48:10 +08:00
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/* 访问队首元素 */
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let peek = queue.first!
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2023-01-19 02:04:21 +08:00
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2023-01-11 23:48:10 +08:00
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/* 元素出队 */
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2023-03-13 21:58:21 +08:00
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// 由于是数组,因此 removeFirst 的复杂度为 O(n)
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2023-01-11 23:48:10 +08:00
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let pool = queue.removeFirst()
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2023-01-19 02:04:21 +08:00
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2023-01-11 23:48:10 +08:00
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/* 获取队列的长度 */
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let size = queue.count
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2023-01-19 02:04:21 +08:00
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2023-01-11 23:48:10 +08:00
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/* 判断队列是否为空 */
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let isEmpty = queue.isEmpty
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2023-01-08 19:41:05 +08:00
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```
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2023-02-01 22:03:04 +08:00
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=== "Zig"
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```zig title="queue.zig"
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```
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2023-02-16 03:39:01 +08:00
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## 队列实现
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2022-11-22 17:47:26 +08:00
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队列需要一种可以在一端添加,并在另一端删除的数据结构,也可以使用链表或数组来实现。
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### 基于链表的实现
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2023-04-09 04:32:17 +08:00
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我们将链表的「头节点」和「尾节点」分别看作是队首和队尾,并规定队尾只可添加节点,队首只可删除节点。
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2022-11-22 17:47:26 +08:00
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2023-01-19 02:04:21 +08:00
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=== "LinkedListQueue"
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2023-02-26 19:22:46 +08:00
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![基于链表实现队列的入队出队操作](queue.assets/linkedlist_queue.png)
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2023-01-19 02:04:21 +08:00
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=== "push()"
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![linkedlist_queue_push](queue.assets/linkedlist_queue_push.png)
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2023-03-13 21:58:21 +08:00
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=== "pop()"
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![linkedlist_queue_pop](queue.assets/linkedlist_queue_pop.png)
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2023-01-19 02:04:21 +08:00
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以下是使用链表实现队列的示例代码。
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2022-11-22 17:47:26 +08:00
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=== "Java"
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```java title="linkedlist_queue.java"
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2023-02-07 04:43:52 +08:00
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[class]{LinkedListQueue}-[func]{}
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2022-11-22 17:47:26 +08:00
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```
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=== "C++"
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```cpp title="linkedlist_queue.cpp"
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2023-02-08 04:17:26 +08:00
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[class]{LinkedListQueue}-[func]{}
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2022-11-22 17:47:26 +08:00
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```
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=== "Python"
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```python title="linkedlist_queue.py"
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2023-02-06 23:23:21 +08:00
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[class]{LinkedListQueue}-[func]{}
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2022-11-22 17:47:26 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "Go"
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```go title="linkedlist_queue.go"
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2023-02-09 04:45:06 +08:00
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[class]{linkedListQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "JavaScript"
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2023-02-08 04:27:55 +08:00
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```javascript title="linkedlist_queue.js"
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2023-02-08 19:45:06 +08:00
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[class]{LinkedListQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "TypeScript"
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```typescript title="linkedlist_queue.ts"
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2023-02-08 19:45:06 +08:00
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[class]{LinkedListQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "C"
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```c title="linkedlist_queue.c"
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2023-02-11 18:22:27 +08:00
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[class]{linkedListQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "C#"
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```csharp title="linkedlist_queue.cs"
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2023-02-08 22:18:02 +08:00
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[class]{LinkedListQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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2023-01-08 19:41:05 +08:00
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=== "Swift"
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```swift title="linkedlist_queue.swift"
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2023-02-08 20:30:05 +08:00
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[class]{LinkedListQueue}-[func]{}
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2023-01-08 19:41:05 +08:00
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```
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2023-02-01 22:03:04 +08:00
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=== "Zig"
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```zig title="linkedlist_queue.zig"
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2023-02-09 22:57:25 +08:00
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[class]{LinkedListQueue}-[func]{}
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2023-02-01 22:03:04 +08:00
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```
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2022-11-22 17:47:26 +08:00
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### 基于数组的实现
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2023-02-16 03:03:02 +08:00
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数组的删除首元素的时间复杂度为 $O(n)$ ,这会导致出队操作效率低下。然而,我们可以采取下述的巧妙方法来避免这个问题。
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考虑借助一个变量 `front` 来指向队首元素的索引,并维护变量 `queSize` 来记录队列长度。我们定义 `rear = front + queSize` ,该公式计算出来的 `rear` 指向“队尾元素索引 $+1$ ”的位置。
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在该设计下,**数组中包含元素的有效区间为 `[front, rear - 1]`** ,进而
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- 对于入队操作,将输入元素赋值给 `rear` 索引处,并将 `queSize` 自增 $1$ 即可;
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- 对于出队操作,仅需将 `front` 自增 $1$ ,并将 `queSize` 自减 $1$ 即可;
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观察发现,入队与出队操作都仅需单次操作即可完成,时间复杂度皆为 $O(1)$ 。
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2022-11-22 17:47:26 +08:00
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2023-01-19 02:04:21 +08:00
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=== "ArrayQueue"
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2023-02-26 19:22:46 +08:00
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![基于数组实现队列的入队出队操作](queue.assets/array_queue.png)
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2023-01-19 02:04:21 +08:00
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=== "push()"
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![array_queue_push](queue.assets/array_queue_push.png)
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2023-03-13 21:58:21 +08:00
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=== "pop()"
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![array_queue_pop](queue.assets/array_queue_pop.png)
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2022-11-22 17:47:26 +08:00
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2023-02-16 03:03:02 +08:00
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细心的同学可能会发现一个问题:在不断入队与出队的过程中,`front` 和 `rear` 都在向右移动,**在到达数组尾部后就无法继续移动了**。为解决此问题,**我们考虑将数组看作是首尾相接的**,这样的数组被称为「环形数组」。
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2022-11-22 17:47:26 +08:00
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2023-02-16 03:03:02 +08:00
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对于环形数组,我们需要令 `front` 或 `rear` 在越过数组尾部后,直接绕回到数组头部接续遍历。这种周期性规律可以通过「取余操作」来实现,详情请见以下代码。
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2022-11-22 17:47:26 +08:00
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=== "Java"
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```java title="array_queue.java"
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2023-02-07 04:43:52 +08:00
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[class]{ArrayQueue}-[func]{}
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2022-11-22 17:47:26 +08:00
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```
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=== "C++"
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```cpp title="array_queue.cpp"
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2023-02-08 04:17:26 +08:00
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[class]{ArrayQueue}-[func]{}
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2022-11-22 17:47:26 +08:00
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```
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=== "Python"
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```python title="array_queue.py"
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2023-02-06 23:23:21 +08:00
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[class]{ArrayQueue}-[func]{}
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2022-11-22 17:47:26 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "Go"
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```go title="array_queue.go"
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2023-02-09 04:45:06 +08:00
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[class]{arrayQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "JavaScript"
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2023-02-08 04:27:55 +08:00
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```javascript title="array_queue.js"
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2023-02-08 19:45:06 +08:00
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[class]{ArrayQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "TypeScript"
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```typescript title="array_queue.ts"
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2023-02-08 19:45:06 +08:00
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[class]{ArrayQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "C"
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```c title="array_queue.c"
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2023-02-11 18:22:27 +08:00
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[class]{arrayQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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=== "C#"
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```csharp title="array_queue.cs"
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2023-02-08 22:18:02 +08:00
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[class]{ArrayQueue}-[func]{}
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2022-12-03 01:31:29 +08:00
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```
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2023-01-08 19:41:05 +08:00
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=== "Swift"
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```swift title="array_queue.swift"
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2023-02-08 20:30:05 +08:00
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[class]{ArrayQueue}-[func]{}
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2023-01-08 19:41:05 +08:00
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```
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2023-02-01 22:03:04 +08:00
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=== "Zig"
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```zig title="array_queue.zig"
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2023-02-09 22:57:25 +08:00
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[class]{ArrayQueue}-[func]{}
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2023-02-01 22:03:04 +08:00
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```
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2023-02-16 03:03:02 +08:00
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以上实现的队列仍存在局限性,即长度不可变。不过这个问题很容易解决,我们可以将数组替换为列表(即动态数组),从而引入扩容机制。有兴趣的同学可以尝试自行实现。
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2023-01-19 02:04:21 +08:00
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2023-02-16 03:39:01 +08:00
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## 两种实现对比
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2023-01-19 02:04:21 +08:00
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与栈的结论一致,在此不再赘述。
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2023-02-16 03:39:01 +08:00
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## 队列典型应用
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2022-11-22 17:47:26 +08:00
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2023-01-07 20:34:32 +08:00
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- **淘宝订单**。购物者下单后,订单就被加入到队列之中,随后系统再根据顺序依次处理队列中的订单。在双十一时,在短时间内会产生海量的订单,如何处理「高并发」则是工程师们需要重点思考的问题。
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2023-02-16 03:03:02 +08:00
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- **各种待办事项**。任何需要实现“先来后到”的功能,例如打印机的任务队列、餐厅的出餐队列等等。
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