2022-11-22 17:47:26 +08:00
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---
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comments: true
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---
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# 冒泡排序
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「冒泡排序 Bubble Sort」是一种最基础的排序算法,非常适合作为第一个学习的排序算法。顾名思义,「冒泡」是该算法的核心操作。
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2022-12-21 01:37:58 +08:00
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!!! question "为什么叫“冒泡”"
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2022-11-22 17:47:26 +08:00
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在水中,越大的泡泡浮力越大,所以最大的泡泡会最先浮到水面。
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「冒泡」操作则是在模拟上述过程,具体做法为:从数组最左端开始向右遍历,依次对比相邻元素大小,若 **左元素 > 右元素** 则将它俩交换,最终可将最大元素移动至数组最右端。
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完成此次冒泡操作后,**数组最大元素已在正确位置,接下来只需排序剩余 $n - 1$ 个元素**。
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=== "Step 1"
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![bubble_operation_step1](bubble_sort.assets/bubble_operation_step1.png)
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=== "Step 2"
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![bubble_operation_step2](bubble_sort.assets/bubble_operation_step2.png)
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=== "Step 3"
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![bubble_operation_step3](bubble_sort.assets/bubble_operation_step3.png)
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=== "Step 4"
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![bubble_operation_step4](bubble_sort.assets/bubble_operation_step4.png)
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=== "Step 5"
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![bubble_operation_step5](bubble_sort.assets/bubble_operation_step5.png)
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=== "Step 6"
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![bubble_operation_step6](bubble_sort.assets/bubble_operation_step6.png)
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=== "Step 7"
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![bubble_operation_step7](bubble_sort.assets/bubble_operation_step7.png)
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2022-11-23 15:50:59 +08:00
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<p align="center"> Fig. 冒泡操作 </p>
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2022-11-22 17:47:26 +08:00
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## 算法流程
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2022-11-23 03:56:25 +08:00
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1. 设数组长度为 $n$ ,完成第一轮「冒泡」后,数组最大元素已在正确位置,接下来只需排序剩余 $n - 1$ 个元素。
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2. 同理,对剩余 $n - 1$ 个元素执行「冒泡」,可将第二大元素交换至正确位置,因而待排序元素只剩 $n - 2$ 个。
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3. 以此类推…… **循环 $n - 1$ 轮「冒泡」,即可完成整个数组的排序**。
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2022-11-22 17:47:26 +08:00
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![bubble_sort](bubble_sort.assets/bubble_sort.png)
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2022-11-23 15:50:59 +08:00
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<p align="center"> Fig. 冒泡排序流程 </p>
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2022-11-22 17:47:26 +08:00
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=== "Java"
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2022-11-27 04:20:30 +08:00
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```java title="bubble_sort.java"
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2022-11-22 17:47:26 +08:00
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/* 冒泡排序 */
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void bubbleSort(int[] nums) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (int i = nums.length - 1; i > 0; i--) {
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// 内循环:冒泡操作
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for (int j = 0; j < i; j++) {
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if (nums[j] > nums[j + 1]) {
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// 交换 nums[j] 与 nums[j + 1]
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int tmp = nums[j];
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nums[j] = nums[j + 1];
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nums[j + 1] = tmp;
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}
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}
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}
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}
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```
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2022-12-01 18:28:57 +08:00
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2022-11-27 04:20:30 +08:00
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=== "C++"
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```cpp title="bubble_sort.cpp"
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/* 冒泡排序 */
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void bubbleSort(vector<int>& nums) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (int i = nums.size() - 1; i > 0; i--) {
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// 内循环:冒泡操作
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for (int j = 0; j < i; j++) {
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if (nums[j] > nums[j + 1]) {
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// 交换 nums[j] 与 nums[j + 1]
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2022-12-27 20:11:22 +08:00
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// 这里使用了 std::swap() 函数
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swap(nums[j], nums[j + 1]);
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2022-11-27 04:20:30 +08:00
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}
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}
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}
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}
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```
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2022-11-22 17:47:26 +08:00
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2022-11-26 01:40:49 +08:00
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=== "Python"
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2022-11-27 04:20:30 +08:00
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```python title="bubble_sort.py"
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2022-11-26 01:40:49 +08:00
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""" 冒泡排序 """
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def bubble_sort(nums):
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n = len(nums)
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# 外循环:待排序元素数量为 n-1, n-2, ..., 1
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2023-01-18 19:09:56 +08:00
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for i in range(n - 1, 0, -1):
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2022-11-26 01:40:49 +08:00
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# 内循环:冒泡操作
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for j in range(i):
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if nums[j] > nums[j + 1]:
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# 交换 nums[j] 与 nums[j + 1]
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nums[j], nums[j + 1] = nums[j + 1], nums[j]
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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="bubble_sort.go"
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2022-12-06 23:41:12 +08:00
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/* 冒泡排序 */
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func bubbleSort(nums []int) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for i := len(nums) - 1; i > 0; i-- {
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// 内循环:冒泡操作
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for j := 0; j < i; j++ {
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if nums[j] > nums[j+1] {
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// 交换 nums[j] 与 nums[j + 1]
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nums[j], nums[j+1] = nums[j+1], nums[j]
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}
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}
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}
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}
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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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```js title="bubble_sort.js"
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/* 冒泡排序 */
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function bubbleSort(nums) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (let i = nums.length - 1; i > 0; i--) {
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// 内循环:冒泡操作
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for (let j = 0; j < i; j++) {
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if (nums[j] > nums[j + 1]) {
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// 交换 nums[j] 与 nums[j + 1]
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let tmp = nums[j];
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nums[j] = nums[j + 1];
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nums[j + 1] = tmp;
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}
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}
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}
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}
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```
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=== "TypeScript"
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```typescript title="bubble_sort.ts"
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2022-12-12 23:18:12 +08:00
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/* 冒泡排序 */
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function bubbleSort(nums: number[]): void {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (let i = nums.length - 1; i > 0; i--) {
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// 内循环:冒泡操作
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for (let j = 0; j < i; j++) {
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if (nums[j] > nums[j + 1]) {
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// 交换 nums[j] 与 nums[j + 1]
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let tmp = nums[j];
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nums[j] = nums[j + 1];
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nums[j + 1] = tmp;
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}
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}
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}
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}
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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="bubble_sort.c"
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2022-12-26 10:17:35 +08:00
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/* 冒泡排序 */
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2023-01-12 00:31:11 +08:00
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void bubbleSort(int nums[], int size) {
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2022-12-26 10:17:35 +08:00
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (int i = 0; i < size - 1; i++)
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{
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// 内循环:冒泡操作
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for (int j = 0; j < size - 1 - i; j++)
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{
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if (nums[j] > nums[j + 1])
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{
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int temp = nums[j];
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nums[j] = nums[j + 1];
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nums[j + 1] = temp;
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}
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}
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}
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}
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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="bubble_sort.cs"
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2022-12-23 15:42:02 +08:00
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/* 冒泡排序 */
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void bubbleSort(int[] nums)
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{
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (int i = nums.Length - 1; i > 0; i--)
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{
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// 内循环:冒泡操作
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for (int j = 0; j < i; j++)
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{
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if (nums[j] > nums[j + 1])
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{
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// 交换 nums[j] 与 nums[j + 1]
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int tmp = nums[j];
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nums[j] = nums[j + 1];
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nums[j + 1] = tmp;
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}
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}
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}
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}
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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="bubble_sort.swift"
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2023-01-31 00:18:40 +08:00
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/* 冒泡排序 */
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func bubbleSort(nums: inout [Int]) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for i in stride(from: nums.count - 1, to: 0, by: -1) {
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// 内循环:冒泡操作
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for j in stride(from: 0, to: i, by: 1) {
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if nums[j] > nums[j + 1] {
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// 交换 nums[j] 与 nums[j + 1]
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let tmp = nums[j]
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nums[j] = nums[j + 1]
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nums[j + 1] = tmp
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}
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}
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}
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}
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2023-01-08 19:41:05 +08:00
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```
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2022-11-23 03:56:25 +08:00
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## 算法特性
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2022-11-22 17:47:26 +08:00
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2023-01-09 22:39:30 +08:00
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**时间复杂度 $O(n^2)$** :各轮「冒泡」遍历的数组长度为 $n - 1$ , $n - 2$ , $\cdots$ , $2$ , $1$ 次,求和为 $\frac{(n - 1) n}{2}$ ,因此使用 $O(n^2)$ 时间。
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2022-11-22 17:47:26 +08:00
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2023-01-09 22:39:30 +08:00
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**空间复杂度 $O(1)$** :指针 $i$ , $j$ 使用常数大小的额外空间。
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2022-11-22 17:47:26 +08:00
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2023-01-09 22:39:30 +08:00
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**原地排序**:指针变量仅使用常数大小额外空间。
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2022-11-22 17:47:26 +08:00
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2023-01-09 22:39:30 +08:00
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**稳定排序**:不交换相等元素。
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2022-11-22 17:47:26 +08:00
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2023-01-16 17:57:28 +08:00
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**自适应排序**:引入 `flag` 优化后(见下文),最佳时间复杂度为 $O(N)$ 。
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2022-11-22 17:47:26 +08:00
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## 效率优化
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我们发现,若在某轮「冒泡」中未执行任何交换操作,则说明数组已经完成排序,可直接返回结果。考虑可以增加一个标志位 `flag` 来监听该情况,若出现则直接返回。
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优化后,冒泡排序的最差和平均时间复杂度仍为 $O(n^2)$ ;而在输入数组 **已排序** 时,达到 **最佳时间复杂度** $O(n)$ 。
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=== "Java"
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2022-11-27 04:20:30 +08:00
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```java title="bubble_sort.java"
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2022-11-22 17:47:26 +08:00
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/* 冒泡排序(标志优化)*/
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void bubbleSortWithFlag(int[] nums) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (int i = nums.length - 1; i > 0; i--) {
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boolean flag = false; // 初始化标志位
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// 内循环:冒泡操作
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for (int j = 0; j < i; j++) {
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if (nums[j] > nums[j + 1]) {
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// 交换 nums[j] 与 nums[j + 1]
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int tmp = nums[j];
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|
nums[j] = nums[j + 1];
|
|
|
|
|
nums[j + 1] = tmp;
|
|
|
|
|
flag = true; // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if (!flag) break; // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
```
|
2022-11-26 01:40:49 +08:00
|
|
|
|
|
2022-11-27 04:20:30 +08:00
|
|
|
|
=== "C++"
|
|
|
|
|
|
|
|
|
|
```cpp title="bubble_sort.cpp"
|
|
|
|
|
/* 冒泡排序(标志优化)*/
|
|
|
|
|
void bubbleSortWithFlag(vector<int>& nums) {
|
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for (int i = nums.size() - 1; i > 0; i--) {
|
|
|
|
|
bool flag = false; // 初始化标志位
|
|
|
|
|
// 内循环:冒泡操作
|
|
|
|
|
for (int j = 0; j < i; j++) {
|
|
|
|
|
if (nums[j] > nums[j + 1]) {
|
|
|
|
|
// 交换 nums[j] 与 nums[j + 1]
|
2022-12-27 20:11:22 +08:00
|
|
|
|
// 这里使用了 std::swap() 函数
|
|
|
|
|
swap(nums[j], nums[j + 1]);
|
2022-11-27 04:20:30 +08:00
|
|
|
|
flag = true; // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if (!flag) break; // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
2022-11-26 01:40:49 +08:00
|
|
|
|
=== "Python"
|
|
|
|
|
|
2022-11-27 04:20:30 +08:00
|
|
|
|
```python title="bubble_sort.py"
|
2022-11-26 01:40:49 +08:00
|
|
|
|
""" 冒泡排序(标志优化) """
|
|
|
|
|
def bubble_sort_with_flag(nums):
|
|
|
|
|
n = len(nums)
|
|
|
|
|
# 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
2023-01-18 19:09:56 +08:00
|
|
|
|
for i in range(n - 1, 0, -1):
|
2022-11-26 01:40:49 +08:00
|
|
|
|
flag = False # 初始化标志位
|
|
|
|
|
# 内循环:冒泡操作
|
|
|
|
|
for j in range(i):
|
|
|
|
|
if nums[j] > nums[j + 1]:
|
|
|
|
|
# 交换 nums[j] 与 nums[j + 1]
|
|
|
|
|
nums[j], nums[j + 1] = nums[j + 1], nums[j]
|
|
|
|
|
flag = True # 记录交换元素
|
|
|
|
|
if not flag:
|
|
|
|
|
break # 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
```
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
|
|
|
|
=== "Go"
|
|
|
|
|
|
|
|
|
|
```go title="bubble_sort.go"
|
2022-12-06 23:41:12 +08:00
|
|
|
|
/* 冒泡排序(标志优化)*/
|
|
|
|
|
func bubbleSortWithFlag(nums []int) {
|
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for i := len(nums) - 1; i > 0; i-- {
|
|
|
|
|
flag := false // 初始化标志位
|
|
|
|
|
// 内循环:冒泡操作
|
|
|
|
|
for j := 0; j < i; j++ {
|
|
|
|
|
if nums[j] > nums[j+1] {
|
|
|
|
|
// 交换 nums[j] 与 nums[j + 1]
|
|
|
|
|
nums[j], nums[j+1] = nums[j+1], nums[j]
|
|
|
|
|
flag = true // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if flag == false { // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
break
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "JavaScript"
|
|
|
|
|
|
|
|
|
|
```js title="bubble_sort.js"
|
|
|
|
|
/* 冒泡排序(标志优化)*/
|
|
|
|
|
function bubbleSortWithFlag(nums) {
|
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for (let i = nums.length - 1; i > 0; i--) {
|
|
|
|
|
let flag = false; // 初始化标志位
|
|
|
|
|
// 内循环:冒泡操作
|
|
|
|
|
for (let j = 0; j < i; j++) {
|
|
|
|
|
if (nums[j] > nums[j + 1]) {
|
|
|
|
|
// 交换 nums[j] 与 nums[j + 1]
|
|
|
|
|
let tmp = nums[j];
|
|
|
|
|
nums[j] = nums[j + 1];
|
|
|
|
|
nums[j + 1] = tmp;
|
|
|
|
|
flag = true; // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if (!flag) break; // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "TypeScript"
|
|
|
|
|
|
|
|
|
|
```typescript title="bubble_sort.ts"
|
2022-12-12 23:18:12 +08:00
|
|
|
|
/* 冒泡排序(标志优化)*/
|
|
|
|
|
function bubbleSortWithFlag(nums: number[]): void {
|
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for (let i = nums.length - 1; i > 0; i--) {
|
|
|
|
|
let flag = false; // 初始化标志位
|
|
|
|
|
// 内循环:冒泡操作
|
|
|
|
|
for (let j = 0; j < i; j++) {
|
|
|
|
|
if (nums[j] > nums[j + 1]) {
|
|
|
|
|
// 交换 nums[j] 与 nums[j + 1]
|
|
|
|
|
let tmp = nums[j];
|
|
|
|
|
nums[j] = nums[j + 1];
|
|
|
|
|
nums[j + 1] = tmp;
|
|
|
|
|
flag = true; // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if (!flag) break; // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
|
|
|
|
```c title="bubble_sort.c"
|
2022-12-26 10:17:35 +08:00
|
|
|
|
/* 冒泡排序 */
|
2023-01-12 00:31:11 +08:00
|
|
|
|
void bubbleSortWithFlag(int nums[], int size) {
|
2022-12-26 10:17:35 +08:00
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for (int i = 0; i < size - 1; i++)
|
|
|
|
|
{
|
|
|
|
|
bool flag = false;
|
|
|
|
|
// 内循环:冒泡操作
|
|
|
|
|
for (int j = 0; j < size - 1 - i; j++)
|
|
|
|
|
{
|
|
|
|
|
if (nums[j] > nums[j + 1])
|
|
|
|
|
{
|
|
|
|
|
int temp = nums[j];
|
|
|
|
|
nums[j] = nums[j + 1];
|
|
|
|
|
nums[j + 1] = temp;
|
|
|
|
|
flag = true;
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-26 12:11:21 +08:00
|
|
|
|
if(!flag) break;
|
2022-12-26 10:17:35 +08:00
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
|
|
|
|
```csharp title="bubble_sort.cs"
|
2022-12-23 15:42:02 +08:00
|
|
|
|
/* 冒泡排序(标志优化)*/
|
|
|
|
|
void bubbleSortWithFlag(int[] nums)
|
|
|
|
|
{
|
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for (int i = nums.Length - 1; i > 0; i--)
|
|
|
|
|
{
|
|
|
|
|
bool flag = false; // 初始化标志位
|
2022-12-24 16:15:41 +08:00
|
|
|
|
// 内循环:冒泡操作
|
2022-12-23 15:42:02 +08:00
|
|
|
|
for (int j = 0; j < i; j++)
|
|
|
|
|
{
|
|
|
|
|
if (nums[j] > nums[j + 1])
|
|
|
|
|
{
|
|
|
|
|
// 交换 nums[j] 与 nums[j + 1]
|
|
|
|
|
int tmp = nums[j];
|
|
|
|
|
nums[j] = nums[j + 1];
|
|
|
|
|
nums[j + 1] = tmp;
|
|
|
|
|
flag = true; // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if (!flag) break; // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
2023-01-08 19:41:05 +08:00
|
|
|
|
|
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title="bubble_sort.swift"
|
2023-01-31 00:18:40 +08:00
|
|
|
|
/* 冒泡排序(标志优化)*/
|
|
|
|
|
func bubbleSortWithFlag(nums: inout [Int]) {
|
|
|
|
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
|
|
|
|
for i in stride(from: nums.count - 1, to: 0, by: -1) {
|
|
|
|
|
var flag = false // 初始化标志位
|
|
|
|
|
for j in stride(from: 0, to: i, by: 1) {
|
|
|
|
|
if nums[j] > nums[j + 1] {
|
|
|
|
|
// 交换 nums[j] 与 nums[j + 1]
|
|
|
|
|
let tmp = nums[j]
|
|
|
|
|
nums[j] = nums[j + 1]
|
|
|
|
|
nums[j + 1] = tmp
|
|
|
|
|
flag = true // 记录交换元素
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
if !flag { // 此轮冒泡未交换任何元素,直接跳出
|
|
|
|
|
break
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
2023-01-08 19:41:05 +08:00
|
|
|
|
```
|