数据结构：10、排序

本文将会介绍8种排序，并在文章末附上代码

一、排序的概念

排序：所谓排序，就是使一串记录，按照其中的某个或某些关键字的大小，递增或递减的排列起来的操作。
稳定性：假定在待排序的记录序列中，存在多个具有相同的关键字的记录，若经过排序，这些记录的相对次
序保持不变，即在原序列中，r[i]=r[j]，且r[i]在r[j]之前，而在排序后的序列中，r[i]仍在r[j]之前，则称这种排
序算法是稳定的；否则称为不稳定的。
内部排序：数据元素全部放在内存中的排序。
外部排序：数据元素太多不能同时放在内存中，根据排序过程的要求不能在内外存之间移动数据的排序。

二、排序的实现

1、直接插入排序

直接插入排序就是判断这个数是否比前面的数小，如果小就放在这个地方，然后一趟一趟的，他的排序在越接近有序就越快，代码和测试结果如下。

void InsertSort(int* a, int n)
{for (int i = 1; i < n; i++){int end = i - 1;int tmp = a[i];while (end>=0){if (a[end] > tmp){a[end + 1] = a[end];end--;}else{break;}}a[end + 1] = tmp;}
}

2、希尔排序

希尔排序就是一个叫希尔的大佬发明出来的，既然直接插入排序越接近有序就越快，那么给他划分成一块一块，对每一块进行排序，然后在进行总体排序，代码和测试结果如下。

void ShellSort(int* a, int n)
{int gap = n;while (gap > 1){gap = gap / 3 + 1;  for (int i = 0; i < n - gap; ++i){int end = i;int tmp = a[end + gap];while (end >= 0){if (a[end] > tmp){a[end + gap] = a[end];end -= gap;}else{break;}}a[end + gap] = tmp;}}
}

3、选择排序

这个选择排序的原理就是选出这组数据的最大或者最小插在最前或者最后，听起来就很慢，所以这里进行了一点点优化，一下排序两个数据，但是依然还是很慢很拉跨，在文章末进行所有测试的比较时可以看出，代码与测试结果如下。

void Swap(int* p1, int* p2)
{int tmp = *p1;*p1 = *p2;*p2 = tmp;
}void SelectSort(int* a, int n)
{int begin = 0, end = n - 1;while (begin < end){int maxi = begin, mini = begin;for (int i = begin; i <= end; i++){if (a[i] > a[maxi]){maxi = i;}if (a[i] < a[mini]){mini = i;}}Swap(&a[begin], &a[mini]);if (begin == maxi){maxi = mini;}Swap(&a[end], &a[maxi]);++begin;--end;}
}

4、堆排序

堆排序首先就是建堆，这里是升序所以就是建大堆，当建堆完成后，交换首尾两个数然后进行调整，这个就是堆排序，代码以及测试结果如下。

void AdjustDown(int* a, int n, int parent)
{int child = parent * 2 + 1;while (child < n){if (child + 1 < n && a[child + 1] > a[child]){++child;}if (a[child] > a[parent]){Swap(&a[child], &a[parent]);parent = child;child = parent * 2 + 1;}else{break;}}
}void HeapSort(int* a, int n)
{for (int i = (n - 1 - 1) / 2; i >= 0; --i){AdjustDown(a, n, i);}int end = n - 1;while (end > 0){Swap(&a[0], &a[end]);AdjustDown(a, end, 0);--end;}
}

5、冒泡排序

冒泡排序，肯定都不陌生，我接触的第一个排序就是他，之前一直认为他最好用，直到接触了本文的其他排序，瞬间不香了，他的原理就是每一趟都依次进行比较交换，代码与测试结果如下。

void BubbleSort(int* a, int n)
{for (int j = 0; j < n; ++j){bool exchange = false;for (int i = 1; i < n - j; i++){if (a[i - 1] > a[i]){int tmp = a[i];a[i] = a[i - 1];a[i - 1] = tmp;exchange = true;}}if (exchange == false){break;}}
}

6、快速排序

快速排序是Hoare于1962年提出的一种二叉树结构的交换排序方法，其基本思想为：任取待排序元素序列中的某元素作为基准值，按照该排序码将待排序集合分割成两子序列，左子序列中所有元素均小于基准值，右子序列中所有元素均大于基准值，然后最左右子序列重复该过程，直到所有元素都排列在相应位置上为止。

有三种划分方式，第一种也是他本人发明的，就是选一个做为key然后左边找比key小的数右边找比key大数，找了停止，然后交换左右两边，直到两边相遇，再把相遇的值和key交换，这样就完成了一次排序，就会形成key左边的比key小，右边的比key大。

第二种就是挖坑法，把key的值保存在临时变量里面，然后形成一个坑，然后右边大的左找到比他的小放到坑里，在找到的地方形成一个坑，左边在走找到大的放到坑里，在挖个坑，直到遇到然后把key放到相遇的这个坑里。

第三种就是前后指针的方法，就是prev每次都需要指向从左到它本身之间最后一个小于基准值的数，如果cur的值大于基准值，这时只让cur++,如果cur指向的位置小于基准值，这时我们让prev++后，判断是否与cur的位置相等，若不相等，则交换cur和prev的值。直到cur>end后，我们再交换prev和基准值，这样基准值的位置也就确定了。

普通的递归快排结束了，就是利用栈中进行排序的方式，方法和上面一样，就是把数据的存储利用栈进行压入与取出，代码与测试结果如下。

int GetMidIndex(int* a, int left, int right)
{int mid = (left + right) / 2;if (a[left] < a[mid]){if (a[mid] < a[right]){return mid;}else if (a[left] < a[right]){return right;}else{return left;}}else {if (a[mid] > a[right]){return mid;}else if (a[left] > a[right]){return right;}else{return left;}}
}int PartSort1(int* a, int left, int right)
{int midi = GetMidIndex(a, left, right);Swap(&a[left], &a[midi]);int keyi = left;while (left < right){while (left < right && a[right] >= a[keyi]){--right;}while (left < right && a[left] <= a[keyi]){++left;}Swap(&a[left], &a[right]);}Swap(&a[keyi], &a[left]);return left;
}int PartSort2(int* a, int left, int right)
{int midi = GetMidIndex(a, left, right);Swap(&a[left], &a[midi]);int key = a[left];int hole = left;while (left < right){while (left < right && a[right] >= key){--right;}a[hole] = a[right];hole = right;while (left < right && a[left] <= key){++left;}a[hole] = a[left];hole = left;}a[hole] = key;return hole;
}int PartSort3(int* a, int left, int right)
{int midi = GetMidIndex(a, left, right);Swap(&a[left], &a[midi]);int prev = left;int cur = left + 1;int keyi = left;while (cur <= right){if (a[cur] < a[keyi] && ++prev != cur){Swap(&a[prev], &a[cur]);}++cur;}Swap(&a[prev], &a[keyi]);keyi = prev;return keyi;
}void QuickSort(int* a, int begin, int end)
{if (begin >= end)return;int keyi = PartSort3(a, begin, end);QuickSort(a, begin, keyi - 1);QuickSort(a, keyi + 1, end);
}void QuickSortNonR(int* a, int begin, int end)
{ST st;StackInit(&st);StackPush(&st, end);StackPush(&st, begin);while (!StackEmpty(&st)){int left = StackTop(&st);StackPop(&st);int right = StackTop(&st);StackPop(&st);int keyi = PartSort1(a, left, right);if (keyi + 1 < right){StackPush(&st, right);StackPush(&st, keyi + 1);}if (left < keyi - 1){StackPush(&st, keyi - 1);StackPush(&st, left);}}StackDestroy(&st);
}

7、归并排序

归并排序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，称为二路归并。归并排序核心步骤就是分解合并，不过在合并时要注意边界的问题，代码实现与测试结果如下。

void MergeSort(int* a, int n)
{int* tmp = (int*)malloc(sizeof(int) * n);_MergeSort(a, 0, n - 1, tmp);free(tmp);
}void MergeSortNonR(int* a, int n)
{int* tmp = (int*)malloc(sizeof(int) * n);int gap = 1;while (gap < n){int j = 0;for (int i = 0; i < n; i += 2 * gap){int begin1 = i, end1 = i + gap - 1;int begin2 = i + gap, end2 = i + 2 * gap - 1;//printf("[%d,%d][%d,%d]\n", begin1, end1, begin2, end2);if (end1 >= n || begin2 >= n){break;}if (end2 >= n){end2 = n - 1;}while (begin1 <= end1 && begin2 <= end2){if (a[begin1] <= a[begin2]){tmp[j++] = a[begin1++];}else{tmp[j++] = a[begin2++];}}while (begin1 <= end1){tmp[j++] = a[begin1++];}while (begin2 <= end2){tmp[j++] = a[begin2++];}memcpy(a + i, tmp + i, sizeof(int) * (end2 - i + 1));}//printf("\n");gap *= 2;}free(tmp);
}

8、计数排序

计数排序的原理就是统计相同元素出现次数，根据统计的结果将序列回收到原来的序列中，他的排序很快，计数排序在数据范围集中时，效率很高，但是适用范围及场景有限，代码与测试结果如下。

void CountSort(int* a, int n)
{int min = a[0], max = a[0];for (int i = 0; i < n; i++){if (a[i] < min){min = a[i];}if (a[i] > max){max = a[i];}}int range = max - min + 1;int* countA = (int*)malloc(sizeof(int) * range);memset(countA, 0, sizeof(int) * range);for (int i = 0; i < n; i++){countA[a[i] - min]++;}int k = 0;for (int j = 0; j < range; j++){while (countA[j]--){a[k++] = j + min;}}
}

9、排序的稳定以及时间复杂度

排序方法	平均情况	最好情况	最坏情况	辅助空间	稳定性
冒泡排序	O(n^2)	O(n)	O(n^2)	O(1)	稳定
选择排序	O(n^2)	O(N^2)	O(n^2)	O(1)	不稳定
直接插入排序	O(n^2)	O(n)	O(n^2)	O(1)	稳定
希尔排序	O(n*longn)~O(n^2)	O(n^1.3)	O(n^2)	O(1)	不稳定
堆排序	O(n*longn)	O(n*longn)	O(n*longn)	O(1)	不稳定
归并排序	O(n*longn)	O(n*longn)	O(n*longn)	O(n)	稳定
快速排序	O(n*longn)	O(n*longn)	O(n^2)	O(logn)~O(n)	不稳定

三、代码

test.c

#define _CRT_SECURE_NO_WARNINGS 1
#include "Sort.h"
#include <time.h>void TestInsertSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));InsertSort(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestShellSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));ShellSort(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestBubbleSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));BubbleSort(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestSelectSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));SelectSort(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestHeapSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));HeapSort(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestQuickSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));//QuickSort(a, 0, sizeof(a) / sizeof(int) - 1);QuickSortNonR(a, 0, sizeof(a) / sizeof(int) - 1);PrintfArry(a, sizeof(a) / sizeof(int));
}void TestMergeSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));//MergeSort(a, sizeof(a) / sizeof(int));MergeSortNonR(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestCountSort()
{int a[] = { 9,5,3,4,1,5,3,6,1,2 };PrintfArry(a, sizeof(a) / sizeof(int));CountSort(a, sizeof(a) / sizeof(int));PrintfArry(a, sizeof(a) / sizeof(int));
}void TestOP()
{srand(time(0));const int N = 100000;int* a1 = (int*)malloc(sizeof(int) * N);int* a2 = (int*)malloc(sizeof(int) * N);int* a3 = (int*)malloc(sizeof(int) * N);int* a4 = (int*)malloc(sizeof(int) * N);int* a5 = (int*)malloc(sizeof(int) * N);int* a6 = (int*)malloc(sizeof(int) * N);int* a7 = (int*)malloc(sizeof(int) * N);int* a8 = (int*)malloc(sizeof(int) * N);for (int i = 0; i < N; ++i){a1[i] = rand();a2[i] = a1[i];a3[i] = a1[i];a4[i] = a1[i];a5[i] = a1[i];a6[i] = a1[i];a7[i] = a1[i];a8[i] = a1[i];}int begin1 = clock();InsertSort(a1, N);int end1 = clock();int begin2 = clock();ShellSort(a2, N);int end2 = clock();int begin3 = clock();BubbleSort(a3, N);int end3 = clock();int begin4 = clock();SelectSort(a4, N);int end4 = clock();int begin5 = clock();HeapSort(a5, N);int end5 = clock();int begin6 = clock();QuickSort(a6, 0, N - 1);int end6 = clock();int begin7 = clock();MergeSort(a7, N);int end7 = clock();int begin8 = clock();CountSort(a8, N);int end8 = clock();printf("InsertSort:%d\n", end1 - begin1);printf("ShellSort:%d\n", end2 - begin2);printf("BubbleSort:%d\n", end3 - begin3);printf("SelcetSort:%d\n", end4 - begin4);printf("HeapSort:%d\n", end5 - begin5);printf("QuickSort:%d\n", end6 - begin6);printf("MergeSort:%d\n", end7 - begin7);printf("CountSort:%d\n", end8 - begin8);free(a1);free(a2);free(a3);free(a4);free(a5);free(a6);free(a7);free(a8);
}int main()
{//TestInsertSort();//TestShellSort();//TestBubbleSort();//TestSelectSort();//TestHeapSort();//TestQuickSort();//TestMergeSort();//TestCountSort();TestOP();return 0;
}

Sort.c

#define _CRT_SECURE_NO_WARNINGS 1
#include "Sort.h"
#include "ST.h"
void PrintfArry(int* a, int n)
{for (int i = 0; i < n; i++){printf("%d ", a[i]);}printf("\n");
}void InsertSort(int* a, int n)
{for (int i = 1; i < n; i++){int end = i - 1;int tmp = a[i];while (end>=0){if (a[end] > tmp){a[end + 1] = a[end];end--;}else{break;}}a[end + 1] = tmp;}
}void ShellSort(int* a, int n)
{int gap = n;while (gap > 1){gap = gap / 3 + 1;  for (int i = 0; i < n - gap; ++i){int end = i;int tmp = a[end + gap];while (end >= 0){if (a[end] > tmp){a[end + gap] = a[end];end -= gap;}else{break;}}a[end + gap] = tmp;}}
}void BubbleSort(int* a, int n)
{for (int j = 0; j < n; ++j){bool exchange = false;for (int i = 1; i < n - j; i++){if (a[i - 1] > a[i]){int tmp = a[i];a[i] = a[i - 1];a[i - 1] = tmp;exchange = true;}}if (exchange == false){break;}}
}void Swap(int* p1, int* p2)
{int tmp = *p1;*p1 = *p2;*p2 = tmp;
}void SelectSort(int* a, int n)
{int begin = 0, end = n - 1;while (begin < end){int maxi = begin, mini = begin;for (int i = begin; i <= end; i++){if (a[i] > a[maxi]){maxi = i;}if (a[i] < a[mini]){mini = i;}}Swap(&a[begin], &a[mini]);if (begin == maxi){maxi = mini;}Swap(&a[end], &a[maxi]);++begin;--end;}
}void AdjustDown(int* a, int n, int parent)
{int child = parent * 2 + 1;while (child < n){if (child + 1 < n && a[child + 1] > a[child]){++child;}if (a[child] > a[parent]){Swap(&a[child], &a[parent]);parent = child;child = parent * 2 + 1;}else{break;}}
}void HeapSort(int* a, int n)
{for (int i = (n - 1 - 1) / 2; i >= 0; --i){AdjustDown(a, n, i);}int end = n - 1;while (end > 0){Swap(&a[0], &a[end]);AdjustDown(a, end, 0);--end;}
}int GetMidIndex(int* a, int left, int right)
{int mid = (left + right) / 2;if (a[left] < a[mid]){if (a[mid] < a[right]){return mid;}else if (a[left] < a[right]){return right;}else{return left;}}else {if (a[mid] > a[right]){return mid;}else if (a[left] > a[right]){return right;}else{return left;}}
}int PartSort1(int* a, int left, int right)
{int midi = GetMidIndex(a, left, right);Swap(&a[left], &a[midi]);int keyi = left;while (left < right){while (left < right && a[right] >= a[keyi]){--right;}while (left < right && a[left] <= a[keyi]){++left;}Swap(&a[left], &a[right]);}Swap(&a[keyi], &a[left]);return left;
}int PartSort2(int* a, int left, int right)
{int midi = GetMidIndex(a, left, right);Swap(&a[left], &a[midi]);int key = a[left];int hole = left;while (left < right){while (left < right && a[right] >= key){--right;}a[hole] = a[right];hole = right;while (left < right && a[left] <= key){++left;}a[hole] = a[left];hole = left;}a[hole] = key;return hole;
}int PartSort3(int* a, int left, int right)
{int midi = GetMidIndex(a, left, right);Swap(&a[left], &a[midi]);int prev = left;int cur = left + 1;int keyi = left;while (cur <= right){if (a[cur] < a[keyi] && ++prev != cur){Swap(&a[prev], &a[cur]);}++cur;}Swap(&a[prev], &a[keyi]);keyi = prev;return keyi;
}void QuickSort(int* a, int begin, int end)
{if (begin >= end)return;int keyi = PartSort3(a, begin, end);QuickSort(a, begin, keyi - 1);QuickSort(a, keyi + 1, end);
}void QuickSortNonR(int* a, int begin, int end)
{ST st;StackInit(&st);StackPush(&st, end);StackPush(&st, begin);while (!StackEmpty(&st)){int left = StackTop(&st);StackPop(&st);int right = StackTop(&st);StackPop(&st);int keyi = PartSort1(a, left, right);if (keyi + 1 < right){StackPush(&st, right);StackPush(&st, keyi + 1);}if (left < keyi - 1){StackPush(&st, keyi - 1);StackPush(&st, left);}}StackDestroy(&st);
}void _MergeSort(int* a, int begin, int end, int* tmp)
{if (begin == end)return;int mid = (begin + end) / 2;_MergeSort(a, begin, mid, tmp);_MergeSort(a, mid + 1, end, tmp);int begin1 = begin, end1 = mid;int begin2 = mid + 1, end2 = end;int i = begin;while (begin1 <= end1 && begin2 <= end2){if (a[begin1] <= a[begin2]){tmp[i++] = a[begin1++];}else{tmp[i++] = a[begin2++];}}while (begin1 <= end1){tmp[i++] = a[begin1++];}while (begin2 <= end2){tmp[i++] = a[begin2++];}memcpy(a + begin, tmp + begin, sizeof(int) * (end - begin + 1));
}void MergeSort(int* a, int n)
{int* tmp = (int*)malloc(sizeof(int) * n);_MergeSort(a, 0, n - 1, tmp);free(tmp);
}void MergeSortNonR(int* a, int n)
{int* tmp = (int*)malloc(sizeof(int) * n);int gap = 1;while (gap < n){int j = 0;for (int i = 0; i < n; i += 2 * gap){int begin1 = i, end1 = i + gap - 1;int begin2 = i + gap, end2 = i + 2 * gap - 1;//printf("[%d,%d][%d,%d]\n", begin1, end1, begin2, end2);if (end1 >= n || begin2 >= n){break;}if (end2 >= n){end2 = n - 1;}while (begin1 <= end1 && begin2 <= end2){if (a[begin1] <= a[begin2]){tmp[j++] = a[begin1++];}else{tmp[j++] = a[begin2++];}}while (begin1 <= end1){tmp[j++] = a[begin1++];}while (begin2 <= end2){tmp[j++] = a[begin2++];}memcpy(a + i, tmp + i, sizeof(int) * (end2 - i + 1));}//printf("\n");gap *= 2;}free(tmp);
}void CountSort(int* a, int n)
{int min = a[0], max = a[0];for (int i = 0; i < n; i++){if (a[i] < min){min = a[i];}if (a[i] > max){max = a[i];}}int range = max - min + 1;int* countA = (int*)malloc(sizeof(int) * range);memset(countA, 0, sizeof(int) * range);for (int i = 0; i < n; i++){countA[a[i] - min]++;}int k = 0;for (int j = 0; j < range; j++){while (countA[j]--){a[k++] = j + min;}}
}

Sort.h

#pragma once
#include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#include <stdbool.h>
void InsertSort(int* a,int n);
void PrintfArry(int* a, int n);
void ShellSort(int* a, int n);
void BubbleSort(int* a, int n);
void Swap(int* p1, int* p2);
void SelectSort(int* a, int n);
void AdjustDown(int* a, int n, int parent);
void HeapSort(int* a, int n);
void QuickSort(int* a, int begin, int end);
int PartSort1(int* a, int left, int right);
int PartSort2(int* a, int left, int right);
int PartSort3(int* a, int left, int right);
void QuickSortNonR(int* a, int begin, int end);
void _MergeSort(int* a, int begin, int end, int* tmp);
void MergeSort(int* a, int n);
void MergeSortNonR(int* a, int n);
void CountSort(int* a, int n);

ST.c

#define _CRT_SECURE_NO_WARNINGS 1#include "ST.h"// 初始化栈
void StackInit(ST* ps)
{assert(ps);ps->a = NULL;ps->capacity = 0;ps->top = 0;
}// 入栈
void StackPush(ST* ps, STDatetype data)
{assert(ps);if (ps->capacity == ps->top){int newcapacity = ps->capacity == 0 ? 4 : ps->capacity * 2;ST* newnode = (ST*)realloc(ps->a, newcapacity * sizeof(STDatetype));if (newnode == NULL){perror("recalloc fail");return;}ps->a = newnode;ps->capacity = newcapacity;}ps->a[ps->top] = data;ps->top++;
}// 出栈
void StackPop(ST* ps)
{assert(ps);assert(!StackEmpty(ps));ps->top--;
}// 获取栈顶元素
STDatetype StackTop(ST* ps)
{assert(ps);assert(!StackEmpty(ps));return ps->a[ps->top-1];
}// 获取栈中有效元素个数
int StackSize(ST* ps)
{assert(ps);assert(!StackEmpty(ps));return ps->top;
}// 检测栈是否为空，如果为空返回非零结果，如果不为空返回0
bool StackEmpty(ST* ps)
{assert(ps);return ps->top == 0;
}// 销毁栈
void StackDestroy(ST* ps)
{assert(ps);free(ps->a);ps->a = NULL;ps->capacity = ps->top = 0;
}

ST.h

#pragma once
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <assert.h>typedef int STDatetype;typedef struct Stack
{STDatetype* a;int top;int capacity;
}ST;// 初始化栈
void StackInit(ST* ps);
// 入栈
void StackPush(ST* ps, STDatetype data);
// 出栈
void StackPop(ST* ps);
// 获取栈顶元素
STDatetype StackTop(ST* ps);
// 获取栈中有效元素个数
int StackSize(ST* ps);
// 检测栈是否为空，如果为空返回非零结果，如果不为空返回0
bool StackEmpty(ST* ps);
// 销毁栈
void StackDestroy(ST* ps);