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。
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；
在《数据结构与算法 JavaScript 描述》中，作者给出了自下而上的迭代方法。但是对于递归法，作者却认为：
However, it is not possible to do so in JavaScript, as the recursion goes too deep for the language to handle.
然而，在 JavaScript 中这种方式不太可行，因为这个算法的递归深度对它来讲太深了。
说实话，我不太理解这句话。意思是 JavaScript 编译器内存太小，递归太深容易造成内存溢出吗？还望有大神能够指教。
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。
2. 算法步骤
-
申请空间，使其大小为两个已经排序序列之和，该空间用来存放合并后的序列；
-
设定两个指针，最初位置分别为两个已经排序序列的起始位置；
-
比较两个指针所指向的元素，选择相对小的元素放入到合并空间，并移动指针到下一位置；
-
重复步骤 3 直到某一指针达到序列尾；
-
将另一序列剩下的所有元素直接复制到合并序列尾。
3. 动图演示
代码实现
JavaScript
实例
function mergeSort(arr) { // 采用自上而下的递归方法
var len = arr.length;
if(len < 2) {
return arr;
}
var middle = Math.floor(len / 2),
left = arr.slice(0, middle),
right = arr.slice(middle);
return merge(mergeSort(left), mergeSort(right));
}
function merge(left, right)
{
var result = [];
while (left.length && right.length) {
if (left[0] <= right[0]) {
result.push(left.shift());
} else {
result.push(right.shift());
}
}
while (left.length)
result.push(left.shift());
while (right.length)
result.push(right.shift());
return result;
}
var len = arr.length;
if(len < 2) {
return arr;
}
var middle = Math.floor(len / 2),
left = arr.slice(0, middle),
right = arr.slice(middle);
return merge(mergeSort(left), mergeSort(right));
}
function merge(left, right)
{
var result = [];
while (left.length && right.length) {
if (left[0] <= right[0]) {
result.push(left.shift());
} else {
result.push(right.shift());
}
}
while (left.length)
result.push(left.shift());
while (right.length)
result.push(right.shift());
return result;
}
Python
实例
def mergeSort(arr):
import math
if(len(arr)<2):
return arr
middle = 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
import math
if(len(arr)<2):
return arr
middle = 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
Go
实例
func mergeSort(arr []int) []int {
length := len(arr)
if length < 2 {
return arr
}
middle := length / 2
left := arr[0:middle]
right := arr[middle:]
return merge(mergeSort(left), mergeSort(right))
}
func merge(left []int, right []int) []int {
var result []int
for 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
}
length := len(arr)
if length < 2 {
return arr
}
middle := length / 2
left := arr[0:middle]
right := arr[middle:]
return merge(mergeSort(left), mergeSort(right))
}
func merge(left []int, right []int) []int {
var result []int
for 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
}
Java
实例
public class MergeSort implements IArraySort {
@Override
public 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;
}
}
@Override
public 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;
}
}
PHP
实例
function mergeSort($arr)
{
$len = count($arr);
if ($len < 2) {
return $arr;
}
$middle = floor($len / 2);
$left = array_slice($arr, 0, $middle);
$right = array_slice($arr, $middle);
return merge(mergeSort($left), mergeSort($right));
}
function merge($left, $right)
{
$result = [];
while (count($left) > 0 && count($right) > 0) {
if ($left[0] <= $right[0]) {
$result[] = array_shift($left);
} else {
$result[] = array_shift($right);
}
}
while (count($left))
$result[] = array_shift($left);
while (count($right))
$result[] = array_shift($right);
return $result;
}
{
$len = count($arr);
if ($len < 2) {
return $arr;
}
$middle = floor($len / 2);
$left = array_slice($arr, 0, $middle);
$right = array_slice($arr, $middle);
return merge(mergeSort($left), mergeSort($right));
}
function merge($left, $right)
{
$result = [];
while (count($left) > 0 && count($right) > 0) {
if ($left[0] <= $right[0]) {
$result[] = array_shift($left);
} else {
$result[] = array_shift($right);
}
}
while (count($left))
$result[] = array_shift($left);
while (count($right))
$result[] = array_shift($right);
return $result;
}
C
实例
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);
}
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);
}
递归版：
实例
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);
}
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);
}
C++
迭代版：
实例
template<typename T> // 整數或浮點數皆可使用,若要使用物件(class)時必須設定"小於"(<)的運算子功能
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;
}
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;
}
递归版：
实例
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 RightSubArray
vector<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);
}
// preconditions:
// Array[front...mid] is sorted
// Array[mid+1 ... end] is sorted
// Copy Array[front ... mid] to LeftSubArray
// Copy Array[mid+1 ... end] to RightSubArray
vector<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);
}
C#
实例
public static List<int> sort(List<int> lst) {
if (lst.Count <= 1)
return lst;
int mid = lst.Count / 2;
List<int> left = new List<int>(); // 定义左侧List
List<int> right = new List<int>(); // 定义右侧List
// 以下兩個循環把 lst 分為左右兩個 List
for (int i = 0; i < mid; i++)
left.Add(lst[i]);
for (int j = mid; j < lst.Count; j++)
right.Add(lst[j]);
left = sort(left);
right = sort(right);
return merge(left, right);
}
/// <summary>
/// 合併兩個已經排好序的List
/// </summary>
/// <param name="left">左側List</param>
/// <param name="right">右側List</param>
/// <returns></returns>
static List<int> merge(List<int> left, List<int> right) {
List<int> temp = new List<int>();
while (left.Count > 0 && right.Count > 0) {
if (left[0] <= right[0]) {
temp.Add(left[0]);
left.RemoveAt(0);
} else {
temp.Add(right[0]);
right.RemoveAt(0);
}
}
if (left.Count > 0) {
for (int i = 0; i < left.Count; i++)
temp.Add(left[i]);
}
if (right.Count > 0) {
for (int i = 0; i < right.Count; i++)
temp.Add(right[i]);
}
return temp;
}
if (lst.Count <= 1)
return lst;
int mid = lst.Count / 2;
List<int> left = new List<int>(); // 定义左侧List
List<int> right = new List<int>(); // 定义右侧List
// 以下兩個循環把 lst 分為左右兩個 List
for (int i = 0; i < mid; i++)
left.Add(lst[i]);
for (int j = mid; j < lst.Count; j++)
right.Add(lst[j]);
left = sort(left);
right = sort(right);
return merge(left, right);
}
/// <summary>
/// 合併兩個已經排好序的List
/// </summary>
/// <param name="left">左側List</param>
/// <param name="right">右側List</param>
/// <returns></returns>
static List<int> merge(List<int> left, List<int> right) {
List<int> temp = new List<int>();
while (left.Count > 0 && right.Count > 0) {
if (left[0] <= right[0]) {
temp.Add(left[0]);
left.RemoveAt(0);
} else {
temp.Add(right[0]);
right.RemoveAt(0);
}
}
if (left.Count > 0) {
for (int i = 0; i < left.Count; i++)
temp.Add(left[i]);
}
if (right.Count > 0) {
for (int i = 0; i < right.Count; i++)
temp.Add(right[i]);
}
return temp;
}
Ruby
实例
def merge list
return list if list.size < 2
pivot = list.size / 2
# Merge
lambda { |left, right|
final = []
until left.empty? or right.empty?
final << if left.first < right.first; left.shift else right.shift end
end
final + left + right
}.call merge(list[0...pivot]), merge(list[pivot..-1])
end
return list if list.size < 2
pivot = list.size / 2
# Merge
lambda { |left, right|
final = []
until left.empty? or right.empty?
final << if left.first < right.first; left.shift else right.shift end
end
final + left + right
}.call merge(list[0...pivot]), merge(list[pivot..-1])
end
参考地址：
https://github.com/hustcc/JS-Sorting-Algorithm/blob/master/5.mergeSort.md
https://zh.wikipedia.org/wiki/%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F
ees
429***0967@qq.com
参考地址
分而治之
可以看到这种结构很像一棵完全二叉树，本文的归并排序我们采用递归去实现（也可采用迭代的方式去实现）。分阶段可以理解为就是递归拆分子序列的过程，递归深度为log2n。
合并相邻有序子序列
再来看看治阶段，我们需要将两个已经有序的子序列合并成一个有序序列，比如上图中的最后一次合并，要将[4,5,7,8]和[1,2,3,6]两个已经有序的子序列，合并为最终序列[1,2,3,4,5,6,7,8]，来看下实现步骤。
import java.util.Arrays; /** * Created by chengxiao on 2016/12/8. */ public class MergeSort { public static void main(String []args){ int []arr = {9,8,7,6,5,4,3,2,1}; sort(arr); System.out.println(Arrays.toString(arr)); } public static void sort(int []arr){ int []temp = new int[arr.length];//在排序前，先建好一个长度等于原数组长度的临时数组，避免递归中频繁开辟空间 sort(arr,0,arr.length-1,temp); } private static void sort(int[] arr,int left,int right,int []temp){ if(left<right){ int mid = (left+right)/2; sort(arr,left,mid,temp);//左边归并排序，使得左子序列有序 sort(arr,mid+1,right,temp);//右边归并排序，使得右子序列有序 merge(arr,left,mid,right,temp);//将两个有序子数组合并操作 } } private static void merge(int[] arr,int left,int mid,int right,int[] temp){ int i = left;//左序列指针 int j = mid+1;//右序列指针 int t = 0;//临时数组指针 while (i<=mid && j<=right){ if(arr[i]<=arr[j]){ temp[t++] = arr[i++]; }else { temp[t++] = arr[j++]; } } while(i<=mid){//将左边剩余元素填充进temp中 temp[t++] = arr[i++]; } while(j<=right){//将右序列剩余元素填充进temp中 temp[t++] = arr[j++]; } t = 0; //将temp中的元素全部拷贝到原数组中 while(left <= right){ arr[left++] = temp[t++]; } } }ees
429***0967@qq.com
参考地址