Problem Statement
Inversion Count for an array indicates – how far (or close) the array is from being sorted. If array is already sorted then inversion count is 0. If array is sorted in reverse order that inversion count is the maximum. Specifically, if there are two numbers
a[i] and
a[j] where
a[i] > a[j] and
i < j it is known as an inversion. You can think of an inversion as a pair of out of order numbers.
Sample Test Cases
Input: arr[] = {8, 4, 2, 1} Output: 6 Explanation: Given array has six inversions: (8,4), (4,2),(8,2), (8,1), (4,1), (2,1). Input: arr[] = {3, 1, 2} Output: 2 Explanation: Given array has two inversions: (3, 1), (3, 2)
Problem Solution
The idea is similar to merge sort, divide the array into two equal or almost equal halves in each step until the base case is reached. a number of inversions, that needs to be added a number of inversions in the left subarray, right subarray and merge().
Create a function merge that counts the number of inversions when two halves of the array are merged, create two indices i and j, i is the index, number of inversion in the second half and the number of inversions by merging the two.
The base case of recursion is when there is only one element in the given half.
Complexity Analysis
Time Complexity: O(n log n), The algorithm used is divide and conquer, So in each level one full array traversal is needed and there are log n levels so the time complexity is O(n log n).
Space Compelxity: O(n), Temporary array.
Code Implementation
#include <bits/stdc++.h> using namespace std; 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[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 =; } /* 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()*/; } // Driver code int main() { int arr[] = { 1, 20, 6, 4, 5 }; int n = sizeof(arr) / sizeof(arr[0]); int ans = mergeSort(arr, n); cout << " Number of inversions are " << ans; return 0; }