Count inversions in a permutation of first N natural numbers

Given an array, arr[] of size N denoting a permutation of numbers from 1 to N, the task is to count the number of inversions in the array.
Note: Two array elements a[i] and a[j] form an inversion if a[i] > a[j] and i < j.

Examples:

Input: arr[] = {2, 3, 1, 5, 4}
Output: 3
Explanation: Given array has 3 inversions: (2, 1), (3, 1), (5, 4).

Input: arr[] = {3, 1, 2}
Output: 2
Explanation: Given array has 2 inversions: (3, 1), (3, 2).

Different methods to solve inversion count has been discussed in the following articles:

Approach: This problem can be solved by using binary search. Follow the steps below to solve the problem:

Store the numbers in the range [1, N] in increasing order in a vector, V.
Initialize a variable, ans as 0 to store the number of inversions in the array, arr[].
Iterate in the range [0, N-1] using the variable i
- Store the index of occurrence of arr[i] in vector V in a variable index.
- Add the count of numbers having positions less than the index that are present in the vector, V since they are smaller than the current element and thus form inversions.
- Remove the element at position index from the vector, V.
Print the value of ans as the result.

Below is the implementation of the above approach:

C++14

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to count number of inversions in
// a permutation of first N natural numbers
int countInversions(int arr[], int n)
{
    vector<int> v;

    // Store array elements in sorted order
    for (int i = 1; i <= n; i++) {
        v.push_back(i);
    }

    // Store the count of inversions
    int ans = 0;

    // Traverse the array
    for (int i = 0; i < n; i++) {

        // Store the index of first
        // occurrence of arr[i] in vector V
        auto itr = lower_bound(
            v.begin(), v.end(), arr[i]);

        // Add count of smaller elements
        // than current element
        ans += itr - v.begin();

        // Erase current element from
        // vector and go to next index
        v.erase(itr);
    }

    // Print the result
    cout << ans;

    return 0;
}

// Driver Code
int main()
{

    // Given Input
    int arr[] = { 2, 3, 1, 5, 4 };
    int n = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    countInversions(arr, n);

    return 0;
}

Java

// Java program for the above approach
import java.util.Vector;

class GFG{

// Function to count number of inversions in
// a permutation of first N natural numbers
static void countInversions(int arr[], int n)
{
    Vector<Integer> v = new Vector<>();

    // Store array elements in sorted order
    for(int i = 1; i <= n; i++)
    {
        v.add(i);
    }

    // Store the count of inversions
    int ans = 0;

    // Traverse the array
    for(int i = 0; i < n; i++) 
    {
        
        // Store the index of first
        // occurrence of arr[i] in vector V
        int itr = v.indexOf(arr[i]);

        // Add count of smaller elements
        // than current element
        ans += itr;

        // Erase current element from
        // vector and go to next index
        v.remove(itr);
    }

    // Print the result
    System.out.println(ans);
}

// Driver code
public static void main(String[] args)
{
    
    // Given Input
    int arr[] = { 2, 3, 1, 5, 4 };
    int n = arr.length;

    // Function Call
    countInversions(arr, n);
}
}

// This code is contributed by abhinavjain194

Python3

# Python3 program for the above approach
from bisect import bisect_left

# Function to count number of inversions in
# a permutation of first N natural numbers
def countInversions(arr, n):
    
    v = []

    # Store array elements in sorted order
    for i in range(1, n + 1, 1):
        v.append(i)

    # Store the count of inversions
    ans = 0

    # Traverse the array
    for i in range(n):
        
        # Store the index of first
        # occurrence of arr[i] in vector V
        itr = bisect_left(v, arr[i])

        # Add count of smaller elements
        # than current element
        ans += itr

        # Erase current element from
        # vector and go to next index
        v = v[:itr] + v[itr + 1 :]

    # Print the result
    print(ans)

# Driver Code
if __name__ == '__main__':
    
    # Given Input
    arr = [ 2, 3, 1, 5, 4 ]
    n = len(arr)

    # Function Call
    countInversions(arr, n)
    
# This code is contributed by SURENDRA_GANGWAR

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {

    // Function to count number of inversions in
    // a permutation of first N natural numbers
    static void countInversions(int[] arr, int n)
    {
        List<int> v = new List<int>();

        // Store array elements in sorted order
        for (int i = 1; i <= n; i++) {
            v.Add(i);
        }

        // Store the count of inversions
        int ans = 0;

        // Traverse the array
        for(int i =0 ;i <n;i ++){

            // Store the index of first
            // occurrence of arr[i] in vector V
            int itr = v.IndexOf(arr[i]);
        

            // Add count of smaller elements
            // than current element
            ans += itr;

            // Erase current element from
            // vector and go to next index
            v.RemoveAt(itr);
        }

        // Print the result
        Console.WriteLine(ans);
    }

    // Driver code
    public static void Main(string[] args)
    {

        // Given Input
        int[] arr = { 2, 3, 1, 5, 4 };
        int n = arr.Length;

        // Function Call
        countInversions(arr, n);
    }
}

// This code is contributed by ukasp.

JavaScript

<script>

// Javascript program for the above approach

// Function to count number of inversions in
// a permutation of first N natural numbers
function countInversions(arr, n)
{
    var v = [];
    var i;
    
    // Store array elements in sorted order
    for(i = 1; i <= n; i++)
    {
        v.push(i);
    }

    // Store the count of inversions
    var ans = 0;

    // Traverse the array
    for(i = 0; i < n; i++) 
    {
        
        // Store the index of first
        // occurrence of arr[i] in vector V
        var index = v.indexOf(arr[i]);

        // Add count of smaller elements
        // than current element
        ans += index;

        // Erase current element from
        // vector and go to next index
        v.splice(index, 1);
    }

    // Print the result
    document.write(ans);
}

// Driver code

// Given Input
var arr = [ 2, 3, 1, 5, 4 ];
var n = arr.length;

// Function Call
countInversions(arr, n);

// This code is contributed by bgangwar59

</script>

Output:

Time Complexity: O(N*log(N))
Auxiliary Space: O(N)

Count inversions in a permutation of first N natural numbers

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