【数据结构】排序算法---归并排序

1. 定义

归并排序（Merge sort）是建立在归并操作上的一种有效的排序算法。该算法是采用分治法（Divide and Conquer）的一个非常典型的应用。将已有序的子序列合并，得到完全有序的序列；即先使每个子序列有序，再使子序列段间有序。若将两个有序表合并成一个有序表，称为2-路归并。

作为一种典型的分而治之思想的算法应用，归并排序的实现由两种方法：

• 自上而下的递归（所有递归的方法都可以用迭代重写，所以就有了第 2 种方法）；
• 自下而上的迭代；

2. 算法步骤

• 把长度为n的输入序列分成两个长度为n/2的子序列；
• 对这两个子序列分别采用归并排序；
• 将两个排序好的子序列合并成一个最终的排序序列。

3. 动图演示

4. 性质

稳定性：

归并排序是高效的基于比较的稳定排序算法。

空间复杂度：

归并排序可以只使用$O(1)$大小的辅助空间，但为便捷通常使用与原数组等长的辅助数组。所以通常情况下空间复杂度为$O(n)$

时间复杂度：

归并排序基于分治思想将数组分段排序后合并，时间复杂度在最优、最坏与平均情况下均为$O(nlogn)$。

5. 算法分析

和选择排序一样，归并排序的性能不受输入数据的影响，但表现比选择排序好的多，因为始终都是$O(nlogn)$的时间复杂度。代价是需要额外的内存空间。

6. 代码实现

C语言——迭代版

int min(int x, int y) {return x < y ? x : y;
}
void merge_sort(int arr[], int len) {int *a = arr;int *b = (int *) malloc(len * sizeof(int));int seg, start;for (seg = 1; seg < len; seg += seg) {for (start = 0; start < len; start += seg * 2) {int low = start, mid = min(start + seg, len), high = min(start + seg * 2, len);int k = low;int start1 = low, end1 = mid;int start2 = mid, end2 = high;while (start1 < end1 && start2 < end2)b[k++] = a[start1] < a[start2] ? a[start1++] : a[start2++];while (start1 < end1)b[k++] = a[start1++];while (start2 < end2)b[k++] = a[start2++];}int *temp = a;a = b;b = temp;}if (a != arr) {int i;for (i = 0; i < len; i++)b[i] = a[i];b = a;}free(b);
}

C语言——递归版

void merge_sort_recursive(int arr[], int reg[], int start, int end) {if (start >= end)return;int len = end - start, mid = (len >> 1) + start;int start1 = start, end1 = mid;int start2 = mid + 1, end2 = end;merge_sort_recursive(arr, reg, start1, end1);merge_sort_recursive(arr, reg, start2, end2);int k = start;while (start1 <= end1 && start2 <= end2)reg[k++] = arr[start1] < arr[start2] ? arr[start1++] : arr[start2++];while (start1 <= end1)reg[k++] = arr[start1++];while (start2 <= end2)reg[k++] = arr[start2++];for (k = start; k <= end; k++)arr[k] = reg[k];
}void merge_sort(int arr[], const int len) {int reg[len];merge_sort_recursive(arr, reg, 0, len - 1);
}

Python

def mergeSort(arr):import mathif(len(arr)<2):return arrmiddle = math.floor(len(arr)/2)left, right = arr[0:middle], arr[middle:]return merge(mergeSort(left), mergeSort(right))def merge(left,right):result = []while left and right:if left[0] <= right[0]:result.append(left.pop(0))else:result.append(right.pop(0));while left:result.append(left.pop(0))while right:result.append(right.pop(0));return result

Java

public class MergeSort implements IArraySort {@Overridepublic int[] sort(int[] sourceArray) throws Exception {// 对 arr 进行拷贝，不改变参数内容int[] arr = Arrays.copyOf(sourceArray, sourceArray.length);if (arr.length < 2) {return arr;}int middle = (int) Math.floor(arr.length / 2);int[] left = Arrays.copyOfRange(arr, 0, middle);int[] right = Arrays.copyOfRange(arr, middle, arr.length);return merge(sort(left), sort(right));}protected int[] merge(int[] left, int[] right) {int[] result = new int[left.length + right.length];int i = 0;while (left.length > 0 && right.length > 0) {if (left[0] <= right[0]) {result[i++] = left[0];left = Arrays.copyOfRange(left, 1, left.length);} else {result[i++] = right[0];right = Arrays.copyOfRange(right, 1, right.length);}}while (left.length > 0) {result[i++] = left[0];left = Arrays.copyOfRange(left, 1, left.length);}while (right.length > 0) {result[i++] = right[0];right = Arrays.copyOfRange(right, 1, right.length);}return result;}}

C++——迭代版

template<typename T> // 整数或浮点数皆可使用,若要使用物件(class)时必须设定"小与"(<)的运算子功能
void merge_sort(T arr[], int len) {T *a = arr;T *b = new T[len];for (int seg = 1; seg < len; seg += seg) {for (int start = 0; start < len; start += seg + seg) {int low = start, mid = min(start + seg, len), high = min(start + seg + seg, len);int k = low;int start1 = low, end1 = mid;int start2 = mid, end2 = high;while (start1 < end1 && start2 < end2)b[k++] = a[start1] < a[start2] ? a[start1++] : a[start2++];while (start1 < end1)b[k++] = a[start1++];while (start2 < end2)b[k++] = a[start2++];}T *temp = a;a = b;b = temp;}if (a != arr) {for (int i = 0; i < len; i++)b[i] = a[i];b = a;}delete[] b;
}

C++——递归版

void Merge(vector<int> &Array, int front, int mid, int end) {// preconditions:// Array[front...mid] is sorted// Array[mid+1 ... end] is sorted// Copy Array[front ... mid] to LeftSubArray// Copy Array[mid+1 ... end] to RightSubArrayvector<int> LeftSubArray(Array.begin() + front, Array.begin() + mid + 1);vector<int> RightSubArray(Array.begin() + mid + 1, Array.begin() + end + 1);int idxLeft = 0, idxRight = 0;LeftSubArray.insert(LeftSubArray.end(), numeric_limits<int>::max());RightSubArray.insert(RightSubArray.end(), numeric_limits<int>::max());// Pick min of LeftSubArray[idxLeft] and RightSubArray[idxRight], and put into Array[i]for (int i = front; i <= end; i++) {if (LeftSubArray[idxLeft] < RightSubArray[idxRight]) {Array[i] = LeftSubArray[idxLeft];idxLeft++;} else {Array[i] = RightSubArray[idxRight];idxRight++;}}
}void MergeSort(vector<int> &Array, int front, int end) {if (front >= end)return;int mid = (front + end) / 2;MergeSort(Array, front, mid);MergeSort(Array, mid + 1, end);Merge(Array, front, mid, end);
}

Go

func mergeSort(arr []int) []int {length := len(arr)if length < 2 {return arr}middle := length / 2left := arr[0:middle]right := arr[middle:]return merge(mergeSort(left), mergeSort(right))
}func merge(left []int, right []int) []int {var result []intfor len(left) != 0 && len(right) != 0 {if left[0] <= right[0] {result = append(result, left[0])left = left[1:]} else {result = append(result, right[0])right = right[1:]}}for len(left) != 0 {result = append(result, left[0])left = left[1:]}for len(right) != 0 {result = append(result, right[0])right = right[1:]}return result
}

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带你初步了解排序算法：https://blog.csdn.net/2301_80191662/article/details/142211265
直接插入排序：https://blog.csdn.net/2301_80191662/article/details/142300973
希尔排序：https://blog.csdn.net/2301_80191662/article/details/142302553
直接选择排序：https://blog.csdn.net/2301_80191662/article/details/142312028
堆排序：https://blog.csdn.net/2301_80191662/article/details/142312338
冒泡排序：https://blog.csdn.net/2301_80191662/article/details/142324131
快速排序：https://blog.csdn.net/2301_80191662/article/details/142324307
归并排序：https://blog.csdn.net/2301_80191662/article/details/142350640
计数排序：https://blog.csdn.net/2301_80191662/article/details/142350741
十大经典排序算法总结与分析：https://blog.csdn.net/2301_80191662/article/details/142211564