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

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++

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// 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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output:

25


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Improved By : ihritik, nidhiva