找出数组中第i小元素

期望时间复杂度：Θ(n)
最坏情况的时间复杂度Θ(n^2)

int randomized_select(int *array,int start,int end,int index){if(start == end)return array[start];int middle = randomized_partition(array,start,end);int position = middle - start + 1;if(index  == position)return array[middle];else if(index < position)return randomized_select(array,start,middle-1,index);elsereturn randomized_select(array,middle + 1,end,index - position);
}

快速排序的随机划分函数:randomized_partition
>>链接地址

本版本为错误版本，中间的中位数求中位数排序用了插入排序，用插入算法，在最坏的情况下，时间复杂度Ω(n^2)

int select(int *array,int start,int end,int index)
{//if(start == end) return array[start];const int constant_num = 5;int remainder = (end - start + 1) % constant_num;int zero = remainder == 0 ? 0 : 1;int num = (end - start + 1) / constant_num + zero;int *median = new int[num];if(remainder){insertion_sort(array,start + (num - 1 ) * constant_num,start + (num - 1 ) * constant_num + remainder-1);median[num-1] = array[start + (num - 1 ) * constant_num + (remainder - 1)/2];}for (int i = 0; i < num - zero; ++i) {insertion_sort(array,start + i * constant_num,start + (i+1) * constant_num - 1);median[i] = array[start + i * constant_num + constant_num / 2];}insertion_sort(median,0,num-1);int key = median[(num-1) / 2];delete[] median;//划分int middle = partition(array,start,end,key);//递归求解int position = middle - start + 1;if(index == position){return  array[middle];}else if(index < position){return select(array,start,middle-1,index);}else{return select(array,middle + 1,end,index - position);}
}

下面是最坏情况的（当n>70时）时间复杂度Θ(n)
以下是正确版本

int select(int *array,int start,int end,int index)
{if(start == end) return array[start];const int constant_num = 5;int remainder = (end - start + 1) % constant_num;int zero = remainder == 0 ? 0 : 1;int num = (end - start + 1) / constant_num + zero;int *median = new int[num];if(remainder){insertion_sort(array,start + (num - 1 ) * constant_num,start + (num - 1 ) * constant_num + remainder-1);median[num-1] = array[start + (num - 1 ) * constant_num + (remainder-1)/2];}for (int i = 0; i < num - zero; ++i) {insertion_sort(array,start + i * constant_num,start + (i+1) * constant_num - 1);median[i] = array[start + i * constant_num + constant_num / 2];}int key = select(median,0,num-1,num/2 + (num % 2 == 0 ? 0 : 1));delete[] median;//划分int middle = partition(array,start,end,key);//递归求解int position = middle - start + 1;if(index == position){return  array[middle];}else if(index < position){return select(array,start,middle-1,index);}else{return select(array,middle + 1,end,index - position);}
}

辅助函数partition

int partition(int *array,int start,int end,int key)
{for (int i = start; i < end + 1; ++i) {if(array[i] == key){int temp = array[i];array[i] = array[end];array[end] = temp;break;}}int middle = start - 1;for (int i = start; i < end; ++i) {if(array[i] <= key){middle++;int temp = array[i];array[i] = array[middle];array[middle] = temp;}}array[end] = array[middle+1];array[middle+1] = key;return middle + 1;
}