用归并排序求逆序数对
假设我们已经知道数组左右两部分的逆序数对(记为inv1和inv2),我们在merge的过程中除了inv1+inv2之外,还需要计算merge过程总的逆序数对。
如何计算merge()?
在归并过程中,让i作为左边数组的遍历索引,j作为右边数组的遍历索引。在合并的过程中,如果a[i]>b[j],那么合并之后就会产生mid-i个逆序数对。因为a[i+1],a[i+2],...,a[mid-1]都会大于b[j].
The complete picture:
Implementation:
#include <bits/stdc++.h> int _mergeSort(int arr[], int temp[], int left, int right); int merge(int arr[], int temp[], int left, int mid, int right); /* This function sorts the input array and returns the number of inversions in the array */ int mergeSort(int arr[], int array_size) { int *temp = (int *)malloc(sizeof(int)*array_size); return _mergeSort(arr, temp, 0, array_size - 1); } /* An auxiliary recursive function that sorts the input array and returns the number of inversions in the array. */ int _mergeSort(int arr[], int temp[], int left, int right) { int mid, inv_count = 0; if (right > left) { /* Divide the array into two parts and call _mergeSortAndCountInv() for each of the parts */ mid = (right + left)/2; /* Inversion count will be sum of inversions in left-part, right-part and number of inversions in merging */ inv_count = _mergeSort(arr, temp, left, mid); inv_count += _mergeSort(arr, temp, mid+1, right); /*Merge the two parts*/ inv_count += merge(arr, temp, left, mid+1, right); } return inv_count; } /* This funt merges two sorted arrays and returns inversion count in the arrays.*/ int merge(int arr[], int temp[], int left, int mid, int right) { int i, j, k; int inv_count = 0; i = left; /* i is index for left subarray*/ j = mid; /* j is index for right subarray*/ k = left; /* k is index for resultant merged subarray*/ while ((i <= mid - 1) && (j <= right)) { if (arr[i] <= arr[j]) { temp[k++] = arr[i++]; } else { temp[k++] = arr[j++]; /*this is tricky -- see above explanation/diagram for merge()*/ inv_count = inv_count + (mid - i); } } /* Copy the remaining elements of left subarray (if there are any) to temp*/ while (i <= mid - 1) temp[k++] = arr[i++]; /* Copy the remaining elements of right subarray (if there are any) to temp*/ while (j <= right) temp[k++] = arr[j++]; /*Copy back the merged elements to original array*/ for (i=left; i <= right; i++) arr[i] = temp[i]; return inv_count; } /* Driver program to test above functions */ int main(int argv, char** args) { int arr[] = {1, 20, 6, 4, 5}; printf(" Number of inversions are %d \n", mergeSort(arr, 5)); getchar(); return 0; }
原文请见:http://www.geeksforgeeks.org/counting-inversions/