像QuickSort一样，合并排序是分而治之的算法。它将输入数组分为两个半部分，将自身称为两个半部分，然后合并两个已排序的半个部分。 merge()函数用于合并两个半部分。 merge(arr，l，m，r)是一个关键过程，假定对arr [l..m]和arr [m + 1..r]进行排序并将两个排序后的子数组合并为一个。有关详细信息，请参见以下C实现。
MergeSort(arr[], l, r)
If r > l
1. Find the middle point to divide the array into two halves:
middle m = l+ (r-l)/2
2. Call mergeSort for first half:
Call mergeSort(arr, l, m)
3. Call mergeSort for second half:
Call mergeSort(arr, m+1, r)
4. Merge the two halves sorted in step 2 and 3:
Call merge(arr, l, m, r)下图来自维基百科，显示了示例数组{38、27、43、3、9、82、10}的完整合并排序过程。如果我们仔细观察一下该图，我们可以看到将数组递归地分成两半，直到大小变为1。一旦大小变为1，合并过程便开始起作用，并开始合并数组，直到完整的数组被合并为止。合并。
C++
// C++ program for Merge Sort
#include
using namespace std;
// Merges two subarrays of arr[].
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r)
{
int n1 = m - l + 1;
int n2 = r - m;
// Create temp arrays
int L[n1], R[n2];
// Copy data to temp arrays L[] and R[]
for (int i = 0; i < n1; i++)
L[i] = arr[l + i];
for (int j = 0; j < n2; j++)
R[j] = arr[m + 1 + j];
// Merge the temp arrays back into arr[l..r]
// Initial index of first subarray
int i = 0;
// Initial index of second subarray
int j = 0;
// Initial index of merged subarray
int k = l;
while (i < n1 && j < n2) {
if (L[i] <= R[j]) {
arr[k] = L[i];
i++;
}
else {
arr[k] = R[j];
j++;
}
k++;
}
// Copy the remaining elements of
// L[], if there are any
while (i < n1) {
arr[k] = L[i];
i++;
k++;
}
// Copy the remaining elements of
// R[], if there are any
while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
// l is for left index and r is
// right index of the sub-array
// of arr to be sorted */
void mergeSort(int arr[],int l,int r){
if(l>=r){
return;//returns recursively
}
int m =l+ (r-l)/2;
mergeSort(arr,l,m);
mergeSort(arr,m+1,r);
merge(arr,l,m,r);
}
// UTILITY FUNCTIONS
// Function to print an array
void printArray(int A[], int size)
{
for (int i = 0; i < size; i++)
cout << A[i] << " ";
}
// Driver code
int main()
{
int arr[] = { 12, 11, 13, 5, 6, 7 };
int arr_size = sizeof(arr) / sizeof(arr[0]);
cout << "Given array is \n";
printArray(arr, arr_size);
mergeSort(arr, 0, arr_size - 1);
cout << "\nSorted array is \n";
printArray(arr, arr_size);
return 0;
}
// This code is contributed by Mayank Tyagi C
/* C program for Merge Sort */
#include
#include
// Merges two subarrays of arr[].
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r)
{
int i, j, k;
int n1 = m - l + 1;
int n2 = r - m;
/* create temp arrays */
int L[n1], R[n2];
/* Copy data to temp arrays L[] and R[] */
for (i = 0; i < n1; i++)
L[i] = arr[l + i];
for (j = 0; j < n2; j++)
R[j] = arr[m + 1 + j];
/* Merge the temp arrays back into arr[l..r]*/
i = 0; // Initial index of first subarray
j = 0; // Initial index of second subarray
k = l; // Initial index of merged subarray
while (i < n1 && j < n2) {
if (L[i] <= R[j]) {
arr[k] = L[i];
i++;
}
else {
arr[k] = R[j];
j++;
}
k++;
}
/* Copy the remaining elements of L[], if there
are any */
while (i < n1) {
arr[k] = L[i];
i++;
k++;
}
/* Copy the remaining elements of R[], if there
are any */
while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
/* l is for left index and r is right index of the
sub-array of arr to be sorted */
void mergeSort(int arr[], int l, int r)
{
if (l < r) {
// Same as (l+r)/2, but avoids overflow for
// large l and h
int m = l + (r - l) / 2;
// Sort first and second halves
mergeSort(arr, l, m);
mergeSort(arr, m + 1, r);
merge(arr, l, m, r);
}
}
/* UTILITY FUNCTIONS */
/* Function to print an array */
void printArray(int A[], int size)
{
int i;
for (i = 0; i < size; i++)
printf("%d ", A[i]);
printf("\n");
}
/* Driver code */
int main()
{
int arr[] = { 12, 11, 13, 5, 6, 7 };
int arr_size = sizeof(arr) / sizeof(arr[0]);
printf("Given array is \n");
printArray(arr, arr_size);
mergeSort(arr, 0, arr_size - 1);
printf("\nSorted array is \n");
printArray(arr, arr_size);
return 0;
} Java
/* Java program for Merge Sort */
class MergeSort
{
// Merges two subarrays of arr[].
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r)
{
// Find sizes of two subarrays to be merged
int n1 = m - l + 1;
int n2 = r - m;
/* Create temp arrays */
int L[] = new int[n1];
int R[] = new int[n2];
/*Copy data to temp arrays*/
for (int i = 0; i < n1; ++i)
L[i] = arr[l + i];
for (int j = 0; j < n2; ++j)
R[j] = arr[m + 1 + j];
/* Merge the temp arrays */
// Initial indexes of first and second subarrays
int i = 0, j = 0;
// Initial index of merged subarry array
int k = l;
while (i < n1 && j < n2) {
if (L[i] <= R[j]) {
arr[k] = L[i];
i++;
}
else {
arr[k] = R[j];
j++;
}
k++;
}
/* Copy remaining elements of L[] if any */
while (i < n1) {
arr[k] = L[i];
i++;
k++;
}
/* Copy remaining elements of R[] if any */
while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
// Main function that sorts arr[l..r] using
// merge()
void sort(int arr[], int l, int r)
{
if (l < r) {
// Find the middle point
int m =l+ (r-l)/2;
// Sort first and second halves
sort(arr, l, m);
sort(arr, m + 1, r);
// Merge the sorted halves
merge(arr, l, m, r);
}
}
/* A utility function to print array of size n */
static void printArray(int arr[])
{
int n = arr.length;
for (int i = 0; i < n; ++i)
System.out.print(arr[i] + " ");
System.out.println();
}
// Driver code
public static void main(String args[])
{
int arr[] = { 12, 11, 13, 5, 6, 7 };
System.out.println("Given Array");
printArray(arr);
MergeSort ob = new MergeSort();
ob.sort(arr, 0, arr.length - 1);
System.out.println("\nSorted array");
printArray(arr);
}
}
/* This code is contributed by Rajat Mishra */Python3
# Python program for implementation of MergeSort
def mergeSort(arr):
if len(arr) > 1:
# Finding the mid of the array
mid = len(arr)//2
# Dividing the array elements
L = arr[:mid]
# into 2 halves
R = arr[mid:]
# Sorting the first half
mergeSort(L)
# Sorting the second half
mergeSort(R)
i = j = k = 0
# Copy data to temp arrays L[] and R[]
while i < len(L) and j < len(R):
if L[i] < R[j]:
arr[k] = L[i]
i += 1
else:
arr[k] = R[j]
j += 1
k += 1
# Checking if any element was left
while i < len(L):
arr[k] = L[i]
i += 1
k += 1
while j < len(R):
arr[k] = R[j]
j += 1
k += 1
# Code to print the list
def printList(arr):
for i in range(len(arr)):
print(arr[i], end=" ")
print()
# Driver Code
if __name__ == '__main__':
arr = [12, 11, 13, 5, 6, 7]
print("Given array is", end="\n")
printList(arr)
mergeSort(arr)
print("Sorted array is: ", end="\n")
printList(arr)
# This code is contributed by Mayank KhannaC#
// C# program for Merge Sort
using System;
class MergeSort {
// Merges two subarrays of []arr.
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int[] arr, int l, int m, int r)
{
// Find sizes of two
// subarrays to be merged
int n1 = m - l + 1;
int n2 = r - m;
// Create temp arrays
int[] L = new int[n1];
int[] R = new int[n2];
int i, j;
// Copy data to temp arrays
for (i = 0; i < n1; ++i)
L[i] = arr[l + i];
for (j = 0; j < n2; ++j)
R[j] = arr[m + 1 + j];
// Merge the temp arrays
// Initial indexes of first
// and second subarrays
i = 0;
j = 0;
// Initial index of merged
// subarry array
int k = l;
while (i < n1 && j < n2) {
if (L[i] <= R[j]) {
arr[k] = L[i];
i++;
}
else {
arr[k] = R[j];
j++;
}
k++;
}
// Copy remaining elements
// of L[] if any
while (i < n1) {
arr[k] = L[i];
i++;
k++;
}
// Copy remaining elements
// of R[] if any
while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
// Main function that
// sorts arr[l..r] using
// merge()
void sort(int[] arr, int l, int r)
{
if (l < r) {
// Find the middle
// point
int m = l+ (r-l)/2;
// Sort first and
// second halves
sort(arr, l, m);
sort(arr, m + 1, r);
// Merge the sorted halves
merge(arr, l, m, r);
}
}
// A utility function to
// print array of size n */
static void printArray(int[] arr)
{
int n = arr.Length;
for (int i = 0; i < n; ++i)
Console.Write(arr[i] + " ");
Console.WriteLine();
}
// Driver code
public static void Main(String[] args)
{
int[] arr = { 12, 11, 13, 5, 6, 7 };
Console.WriteLine("Given Array");
printArray(arr);
MergeSort ob = new MergeSort();
ob.sort(arr, 0, arr.Length - 1);
Console.WriteLine("\nSorted array");
printArray(arr);
}
}
// This code is contributed by Princi Singh输出
Given array is
12 11 13 5 6 7
Sorted array is
5 6 7 11 12 13时间复杂度：在不同计算机上对数组进行排序。合并排序是一种递归算法，时间复杂度可以表示为以下递归关系。
T(n)= 2T(n / 2)+θ(n)
可以使用“递归树”方法或“主控”方法解决以上递归问题。它属于主方法的情况II，并且递归的解是θ(nLogn)。在所有3种情况下(最差，平均和最佳)，合并排序的时间复杂度均为θ(nLogn)，因为合并排序始终将数组分为两半，并花费线性时间来合并两半。
辅助空间： O(n)
算法范式：分而治之
就地排序：在典型实施中为否
稳定：是的
合并排序的应用
- 合并排序对于在O(nLogn)时间内对链表进行排序很有用。在链表的情况下，情况有所不同，主要是由于数组和链表的内存分配不同。与数组不同，链接列表节点在内存中可能不相邻。与数组不同，在链表中，我们可以在O(1)额外空间和O(1)时间的中间插入项目。因此，可以实现合并排序的合并操作而无需为链接列表提供额外的空间。
在数组中，由于元素在内存中是连续的，因此我们可以进行随机访问。假设我们有一个整数(4字节)数组A，并且将A [0]的地址设为x，然后访问A [i]，我们可以直接访问(x + i * 4)处的内存。与数组不同，我们不能在链表中进行随机访问。快速排序需要很多此类访问权限。在访问第i个索引的链表中，我们没有从头到第i个节点的每个节点，因为我们没有连续的内存块。因此，快速排序的开销增加了。合并排序按顺序访问数据，并且对随机访问的需求低。 - 倒数问题
- 用于外部排序
合并排序的缺点
- 与其他排序算法相比，速度较慢，可以完成较小的任务。
- 合并排序算法需要临时数组的额外存储空间0(n)。
- 即使数组已排序，它也要经历整个过程。
- 关于合并排序的最新文章
- 排序的编码实践。
- 合并排序测验