Proizvolov’s Identity

Given two arrays A and B of size N. Array A is in increasing order and B is in decreasing order. Both arrays are the subsequences of the numbers from 1 to 2N. The task is to find the sum of the absolute difference of two arrays.

Sum = |A₁ – B₁| + |A₂ – B₂| + |A₃ – B₃| + …. + |A_N – B_N| 

Examples:

Input : A[] = {1, 2, 3, 4, 5}, B[] = {10, 9, 8, 7, 6} 
Output : 25
Input : A[] = {1, 5, 6, 8, 10, 12}, B[] = {11, 9, 7, 4, 3, 2} 
Output : 36 

Naive Approach: A naive approach is to run a loop and find the sum of the absolute differences.
Efficient Approach: Proizvolov’s identity is an identity concerning sums of the differences of positive integers. It states that if we take first 2N integers and partition them into two subsets of N numbers each. 
Arrange one subset in increasing order : A₁ < A₂ < A₃ < …. < A_N
Arrange another subset in decreasing order : B₁ > B₂ > B₃ > …. > B_N
Then the sum |A₁ – B₁| + |A₂ – B₂| + |A₃ – B₃| + …. + |A_N – B_N| is always equals to N²
Below is the implementation of the above approach: 

C++

// CPP program to implement proizvolov's identity
#include<bits/stdc++.h>
using namespace std;

// Function to implement proizvolov's identity
int proizvolov(int a[], int b[], int n)
{
    // According to proizvolov's identity
    return n*n;
}

// Driver code
int main()
{
    int a[] = {1, 5, 6, 8, 10}, b[] = {9, 7, 4, 3, 2};

    int n = sizeof(a) / sizeof(a[0]);

    // Function call
    cout << proizvolov(a, b, n);

    return 0;
}

Java

// Java program to implement proizvolov's identity
class GFG
{
    // Function to implement proizvolov's identity
    static int proizvolov(int a [], int b [], int n)
    {
        // According to proizvolov's identity
        return n * n;
    }

    // Driver code
    public static void main (String[] args)
    {
        int a [] = {1, 5, 6, 8, 10};
        int b [] = {9, 7, 4, 3, 2};

        int n = a.length;

        // Function call
        System.out.println(proizvolov(a, b, n));
    }
}

// This code is contributed by ihritik

Python3

# Python3 program to implement
# proizvolov's identity

# Function to implement
# proizvolov's identity
def proizvolov(a, b, n):
    return n * n

# Driver code
a = [ 1, 5, 6, 8, 10 ]
b = [ 9, 7, 4, 3, 2 ]
n = len(a)

# Function call
print(proizvolov(a, b, n, ))

# This code is contributed by nidhiva

C#

// C# program to implement proizvolov's identity
using System;

class GFG
{
    // Function to implement proizvolov's identity
    static int proizvolov(int [] a,
                          int [] b, int n)
    {
        // According to proizvolov's identity
        return n * n;
    }

    // Driver code
    public static void Main ()
    {
        int [] a = {1, 5, 6, 8, 10};
        int [] b = {9, 7, 4, 3, 2};

        int n = a.Length;

        // Function call
        Console.WriteLine(proizvolov(a, b, n));
    }
}

// This code is contributed by ihritik

Javascript

<script>

// Javascript program to implement
// proizvolov's identity

// Function to implement proizvolov's identity
function proizvolov(a, b, n)
{
    // According to proizvolov's identity
    return n*n;
}

// Driver code
    let a = [1, 5, 6, 8, 10], b = [9, 7, 4, 3, 2];

    let n = a.length;

    // Function call
    document.write(proizvolov(a, b, n));

</script>

Output:

25

Time complexity: O(1) because constant operations are done

Auxiliary Space: O(1)