分治算法三（随机化快速排序）

1、快速排序对于输入数据的顺序比较敏感。主要在于每次划分对主元的选择。基本的快速排序选取第一个元素作为主元。这样在数组已经有序的情况下，每次划分将得到最坏的结果。这个时候，时间复杂度将会退化到O(n^2)。
2、一种比较常见的优化方法是随机化算法，即随机选取一个元素作为主元。这种情况下虽然最坏情况仍然是O(n^2)，但最坏情况不再依赖于输入数据，而是取决于随机函数随机数的选取。实际上，随机化快速排序得到理论最坏情况的可能性仅为1/(2^n）。所以随机化快速排序可以对于绝大多数输入数据达到O(n*logn）的期望时间复杂度。
3、随机化快速排序的唯一缺点在于，一旦输入数据中有很多的相同数据，随机化的效果将直接减弱。对于极限情况，即对于n个相同的数排序，随机化快速排序的时间复杂度将毫无疑问的降低到O(n^2）。解决方法是用一种方法进行扫描，使没有交换的情况下主元保留在原位置。

/* file: random_quick_sort*//* 1、find the pivot of the Array by using   random algorithm*//* 2、divide the Array into two subarrays*//* 3、conquer the two subarrays*//* 4、the Array is sorted,when conquer ended*/#include<stdio.h>#include <stdlib.h>/*=====================================  swap:swap two numbers a and b=======================================*/inline void swap(int* a,int* b){int tmp;tmp  = *a;*a   = *b;*b   = tmp;}/*=====================================  Partition:Partition the Array from startto end into two subarrays.Return the pivot of the elementArray[start]=======================================*/int Partition(int* Array, int start, int end){//choose Array[start] as the pivot element //divide the array into two subarrays.//left of the Array's elements <= Array[start]//right of the array's elements > Array[start]int pivot = start,j;for(j = start + 1; j <= end; j++)if(Array[j] <= Array[start]){pivot++;swap(&Array[pivot], &Array[j]);}swap(&Array[start], &Array[pivot]);return pivot;}/*=====================================  RandomPartition:using random algorithm.partition the  Array into two subarrays.Return the   pivot of the element Array[start] which  is already swaped=======================================*/int RandomPartition(int* Array, int start, int end){//generate the random index of the pivotint random_index = start + rand() % (end - start + 1);//swap the pivot with the random_index elementswap(&Array[start], &Array[random_index]);//now random_index element act as the start elementreturn Partition(Array,start,end);}/*=====================================  RandomQuickSort:Sort the Array using QuickSort  algorithm.By partition an Array  into two subarrays and then   conquer the two subarrays.=======================================*/void RandomQuickSort(int* Array, int start,int end){int pivot;if(start < end){//find the pivot of the arraypivot = RandomPartition(Array, start, end);//conquer the left subarrayRandomQuickSort(Array, start, pivot - 1);//conquer the right subarrayRandomQuickSort(Array, pivot + 1, end);}}void main(){int n,i;printf("Please input the length of Array<0:exit>:\n");while(scanf("%d",&n),n){//create an arrays of length nint* Array = new int[n];//get the input integersprintf("Please input %d integers:\n",n);for(i = 0; i < n; i++)scanf("%d",&Array[i]);//use QuickSort algorithomRandomQuickSort(Array,0,n-1);//print the sorted arrayprintf("Sorted Array:\n");for(i = 0; i < n; i++)printf("%d ",Array[i]);system("pause");system("cls");//delete the arraydelete[] Array;printf("Please input the length of Array<0:exit>:\n");}}