归并排序

归并排序（英語：Merge sort，或mergesort），是建立在归并操作上的一种有效的排序算法，效率為${\displaystyle O(n\log n)}$（大O符号）。1945年由约翰·冯·诺伊曼首次提出。该算法是采用分治法（Divide and Conquer）的一个非常典型的应用，且各层分治递归可以同时进行。

归并排序
使用合併排序為一列數字進行排序的過程
概况
類別	排序算法
資料結構	数组
复杂度
平均時間複雜度	${\displaystyle \Theta (n\log n)}$
最坏时间复杂度	${\displaystyle \Theta (n\log n)}$
最优时间复杂度	${\displaystyle \Theta (n\log n)}$
空間複雜度	${\displaystyle \Theta (n)}$
最佳解	有时是
相关变量的定义

使用合併排序為一列數字進行排序的過程

概述

采用分治法:

• 分割：递归地把当前序列平均分割成两半。
• 整合：在保持元素顺序的同时将上一步得到的子序列整合到一起（归并）。

归并操作

归并操作（merge），也叫归并算法，指的是将两个已经排序的序列合并成一个序列的操作。归并排序算法依赖归并操作。

递归法（Top-down）

1. 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
2. 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
3. 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
4. 重复步骤3直到某一指针到达序列尾
5. 将另一序列剩下的所有元素直接复制到合并序列尾

迭代法（Bottom-up）

原理如下（假设序列共有${\displaystyle n}$ 个元素）：

1. 将序列每相邻两个数字进行归并操作，形成${\displaystyle ceil(n/2)}$ 个序列，排序后每个序列包含两/一个元素
2. 若此时序列数不是1个则将上述序列再次归并，形成${\displaystyle ceil(n/4)}$ 个序列，每个序列包含四/三个元素
3. 重复步骤2，直到所有元素排序完毕，即序列数为1

實作範例

C語言

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); } 

遞歸版：

// 分治-治 void mergeSort_conquer(int* array, int left, int mid, int right, int* temp) {  // [left, mid]和[mid+1, right]两个有序数组  int i = left;  int j = mid + 1;  int index = 0;  while (i <= mid && j <= right) {  if (array[i] < array[j]) {  temp[index++] = array[i++];  } else {  temp[index++] = array[j++];  }  }  // 剩余元素直接放入temp  while (i <= mid) {  temp[index++] = array[i++];  }  while (j <= right) {  temp[index++] = array[j++];  }  // 放回原数组  index = 0;  while (left <= right) {  array[left++] = temp[index++];  } } // 分治-分 void mergeSort_divide(int* array, int left, int right, int* temp) {  if (left < right) {  int mid = left + (right - left) / 2;  // 左边归并排序  mergeSort_divide(array, left, mid, temp);  // 右边归并排序  mergeSort_divide(array, mid + 1, right, temp);  // 合并两个有序序列  mergeSort_conquer(array, left, mid, right, temp);  } } void mergeSort(int* array, int size) {  int* temp = (int*)malloc(sizeof(int) * size);  mergeSort_divide(array, 0, size - 1, temp); } 

C++

迭代版：

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; } 

遞歸版：

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 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 - front) / 2;  MergeSort(Array, front, mid);  MergeSort(Array, mid + 1, end);  Merge(Array, front, mid, end); } 

^[1]

C#

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; } 

Ruby

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 

Java

遞歸版：

static void merge_sort_recursive(int[] arr, int[] result, 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, result, start1, end1); merge_sort_recursive(arr, result, start2, end2); int k = start; while (start1 <= end1 && start2 <= end2) result[k++] = arr[start1] < arr[start2] ? arr[start1++] : arr[start2++]; while (start1 <= end1) result[k++] = arr[start1++]; while (start2 <= end2) result[k++] = arr[start2++]; for (k = start; k <= end; k++) arr[k] = result[k]; } public static void merge_sort(int[] arr) { int len = arr.length; int[] result = new int[len]; merge_sort_recursive(arr, result, 0, len - 1); } 

迭代版：

public static void merge_sort(int[] arr) { int[] orderedArr = new int[arr.length]; for (int i = 2; i < arr.length * 2; i *= 2) { for (int j = 0; j < (arr.length + i - 1) / i; j++) { int left = i * j; int mid = left + i / 2 >= arr.length ? (arr.length - 1) : (left + i / 2); int right = i * (j + 1) - 1 >= arr.length ? (arr.length - 1) : (i * (j + 1) - 1); int start = left, l = left, m = mid; while (l < mid && m <= right) { if (arr[l] < arr[m]) { orderedArr[start++] = arr[l++]; } else { orderedArr[start++] = arr[m++]; } } while (l < mid) orderedArr[start++] = arr[l++]; while (m <= right) orderedArr[start++] = arr[m++]; System.arraycopy(orderedArr, left, arr, left, right - left + 1); } } } 

PHP

function merge_sort($arr) { $len = count($arr); if ($len <= 1) return $arr; $half = ($len>>1) + ($len & 1); $arr2d = array_chunk($arr, $half); $left = merge_sort($arr2d[0]); $right = merge_sort($arr2d[1]); while (count($left) && count($right)) if ($left[0] < $right[0]) $reg[] = array_shift($left); else $reg[] = array_shift($right); return array_merge($reg, $left, $right); } $arr = array(21, 34, 3, 32, 82, 55, 89, 50, 37, 5, 64, 35, 9, 70); $arr = merge_sort($arr); for ($i = 0; $i < count($arr); $i++) { echo $arr[$i] . ' '; } 

Python3

def mergeSort(nums): if len(nums) < 2: return nums mid = len(nums) // 2 left = mergeSort(nums[:mid]) right = mergeSort(nums[mid:]) i = j = 0 result = [] while i < len(left) and j < len(right): if left[i] < right[j]: result.append(left[i]) i += 1 else: result.append(right[j]) j += 1 while i < len(left): result.append(left[i]) i += 1 while j < len(right): result.append(right[j]) j += 1 return result if __name__ == "__main__": nums = [1, 4, 2, 3.6, -1, 0, 25, -34, 8, 9, 1, 0] print("original:", nums) print("Sorted:", mergeSort(nums)) 

Erlang

%% @doc 归并排序 g_sort([]) ->  []; g_sort([T]) ->  [T]; g_sort(L) ->  g_sort(L, length(L)). g_sort([_, _ | _] = L, Length) ->  SplitNum = trunc(Length / 2),  {L1, L2} = lists:split(SplitNum, L),  g_merge(g_sort(L1, SplitNum), g_sort(L2, Length - SplitNum)); g_sort(L, _Length) ->  L. %% 已经排好序的两个list合并 g_merge([], L2) ->  L2; g_merge(L1, []) ->  L1; g_merge([T1 | Rest1] = L1, [T2 | Rest2] = L2) ->  if  T1 =< T2 -> [T1 | g_merge(Rest1, L2)];  true -> [T2 | g_merge(L1, Rest2)]  end. 

Javascript

递归法

function merge(left, right){ var result = []; while(left.length > 0 && right.length > 0){ if(left[0] < right[0]){ result.push(left.shift()); }else{ result.push(right.shift()); } } return result.concat(left, right); } function mergeSort(arr){ if(arr.length <=1) return arr; var middle = Math.floor(arr.length / 2); var left = arr.slice(0, middle); var right = arr.slice(middle); return merge(mergeSort(left), mergeSort(right)); } 

迭代法

Go

package main import (  "fmt"  "sort" ) func MergeSort(list []int) []int {  var length = len(list)  if length < 2 {  return list  }  var mid = length / 2  return merge(MergeSort(list[:mid]), MergeSort(list[mid:])) } func merge(x, y []int) []int {  var r []int = make([]int, len(x)+len(y))  for i, j := 0, 0; ; {  if i < len(x) && (j == len(y) || x[i] < y[j]) {  r[i+j] = x[i]  i++  } else if j < len(y) {  r[i+j] = y[j]  j++  } else {  break  }  }  return r } func main() {  var list []int = []int{56, 48, 58, 94, 87, 4, 5, 61, 5, 8, 74, 9, 84, 15, 94, 9, 4, 31, 41, 68, 7, 4, 6, 94, 16, 9, 8, 4}  fmt.Println(MergeSort(list))  fmt.Println(list)  sort.Ints(list)  fmt.Println(list) } 

递归版

package main import (  "fmt" ) func merge(data []int) []int {  sum := len(data)  if sum <= 1 {  return data  }  left := data[0 : sum/2]  lSize := len(left)  if lSize >= 2 {  left = merge(left)  }  right := data[sum/2:]  rSize := len(right)  if rSize >= 2 {  right = merge(right)  }  j := 0  t := 0  arr := make([]int, sum)  fmt.Println(left, right, data)  for i := 0; i < sum; i++ {  if j < lSize && t < rSize {  if left[j] <= right[t] {  arr[i] = left[j]  j++  } else {  arr[i] = right[t]  t++  }   } else if j >= lSize{  arr[i] = right[t]  t++  } else if t >= rSize{  arr[i] = left[j]  j++  }  }  return arr } func main() {  var aa = []int{1000, 2, 31, 34, 5, 9, 7, 4, 6, 89, 90, 99, 99, 99, 99, 99}    var bb = merge(aa)  fmt.Println(bb) } 

算法复杂度

比较操作的次数介于${\displaystyle (n\log n)/2}$ 和${\displaystyle n\log n-n+1}$ 。 赋值操作的次数是${\displaystyle (2n\log n)}$ 。归并算法的空间复杂度为：${\displaystyle \Theta (n)}$ 

参考文献

1. Cormen, Thomas H. （英语：Thomas H. Cormen）; Leiserson, Charles E. （英语：Charles E. Leiserson）; Rivest, Ronald L.; Stein, Clifford. Section 2.3: Designing algorithms. Introduction to Algorithms 3rd. MIT Press and McGraw-Hill. 2009: 29–34 [1990]. ISBN 0-262-03384-4. .

外部連結

• Dictionary of Algorithms and Data Structures: Merge sort （页面存档备份，存于互联网档案馆）
• Merge sort Algorithm with example program （页面存档备份，存于互联网档案馆）
• Mergesort For Linked Lists （页面存档备份，存于互联网档案馆）