# 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| + …. + |AN – BN|

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

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive Approach: A navie 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 < …. < AN

Arrange another subset in decreasing order : B1 > B2 > B3 > …. > BN

Then the sum |A1 – B1| + |A2 – B2| + |A3 – B3| + …. + |AN – BN| is always equals to N2

Below is the implementation of the above approach:

## C++

 `// CPP program to implement proizvolov's identity ` `#include ` `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); ` `     `  `    ``// 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 `

Output:

```25
```

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Improved By : ihritik, nidhiva