Maximum Sum of Products of Two Arrays

Given two arrays A and B of positive integers of the same size N. The task is to find the maximum sum of products of their elements. Each element in A has to be multiplied with exactly one element in B or vice versa such that each element of both the arrays appears exactly once and the sum of the product obtained is maximum. 

Examples: 

Input : A[] = {1, 2, 3},  B[] = {4, 5, 1} 
Output : 24 
Explanation : Maximum sum of product is obtained by 5*3+4*2+1*1 = 24.

Input : A[] = {5, 1, 3, 4, 2}, B[] = {8, 10, 9, 7, 6} 
Output : 130 
Explanation : Maximum sum of product is obtained by 10*5+9*4+8*3+7*2+6*1 = 130. 

The idea is to observe that product of two maximum number will contribute toward the maximum sum of the product. So the idea is to: 

1. Sort both the arrays.
2. Traverse the arrays, and calculate the sum of products of array elements that are at the same index.

Implementation:

C++

// CPP program to calculate maximum sum
// of products of two arrays
#include<bits/stdc++.h>
using namespace std;

    // Function that calculates maximum sum
    // of products of two arrays
    int maximumSOP(int *a, int *b)
    {
        // Variable to store the sum of
        // products of array elements
        int sop = 0;

        // length of the arrays
        int n = sizeof(a)/sizeof(a[0]);

        // Sorting both the arrays
        sort(a,a+n+1);
        sort(b,b+n+1);

        // Traversing both the arrays
        // and calculating sum of product
        for (int i = 0; i <=n; i++) {
            sop += a[i] * b[i];
        }

        return sop;
    }

    // Driver code
    int main()
    {
        int A[] = { 1, 2, 3 };
        int B[] = { 4, 5, 1 };

        cout<<maximumSOP(A, B);
        return 0;
    }

// This code is contributed by mits

Java

// Java program to calculate maximum sum
// of products of two arrays

import java.io.*;
import java.util.*;
public class GFG {

    // Function that calculates maximum sum
    // of products of two arrays
    static int maximumSOP(int[] a, int[] b)
    {
        // Variable to store the sum of
        // products of array elements
        int sop = 0;

        // length of the arrays
        int n = a.length;

        // Sorting both the arrays
        Arrays.sort(a);
        Arrays.sort(b);

        // Traversing both the arrays
        // and calculating sum of product
        for (int i = 0; i < n; i++) {
            sop += a[i] * b[i];
        }

        return sop;
    }

    // Driver code
    public static void main(String args[])
    {
        int[] A = { 1, 2, 3 };
        int[] B = { 4, 5, 1 };

        System.out.println(maximumSOP(A, B));
    }
}

Python 3

# Python program to calculate
# maximum sum of products of
# two arrays

# Function that calculates
# maximum sum of products
# of two arrays
def maximumSOP(a, b) :

    # Variable to store the sum of
    # products of array elements
    sop = 0

    # length of the arrays
    n = len(a)

    # Sorting both the arrays
    a.sort()
    b.sort()

    # Traversing both the arrays
    # and calculating sum of product
    for i in range(n) :
        sop += a[i] * b[i]

    return sop

# Driver code
if __name__ == "__main__" :

    A = [1, 2, 3]
    B = [4, 5, 1]

    print(maximumSOP(A, B))

# This code is contributed by ANKITRAI1

C#

using System;

class Program
{
  static int MaximumSOP(int[] a, int[] b)
  {
    // Variable to store the sum of
    // products of array elements
    int sop = 0;

    // length of the arrays
    int n = a.Length;

    // Sorting both the arrays
    Array.Sort(a);
    Array.Sort(b);

    // Traversing both the arrays
    // and calculating sum of product
    for (int i = 0; i < n; i++)
    {
      sop += a[i] * b[i];
    }

    return sop;
  }

  static void Main(string[] args)
  {
    int[] A = { 1, 2, 3 };
    int[] B = { 4, 5, 1 };

    Console.WriteLine(MaximumSOP(A, B));
  }
}

// This code is contributed by shivhack999

PHP

<?php
// PHP program to calculate maximum
// sum of products of two arrays

// Function that calculates maximum
// sum of products of two arrays
function maximumSOP(&$a, &$b)
{
    $sop = 0;

    // Sorting both the arrays
    sort($a);
    sort($b);

    // length of the arrays
    $n = sizeof($a);

    // Traversing both the arrays
    // and calculating sum of product
    for ($i = 0; $i < $n; $i++)
    {
        $sop = $sop + ($a[$i] * $b[$i]);
    }

    return $sop;
}

// Driver code
$A = array(1, 2, 3 );
$B = array(4, 5, 1 );
echo maximumSOP($A, $B);

// This code is contributed
// by Shivi_Aggarwal
?>

Javascript

<script>

// Javascript program to calculate maximum sum
// of products of two arrays

    // Function that calculates maximum sum
    // of products of two arrays
    function maximumSOP(a, b)
    {
        // Variable to store the sum of
        // products of array elements
        let sop = 0;

        // length of the arrays
        let n = a.length;

        // Sorting both the arrays
        a.sort();
        b.sort();

        // Traversing both the arrays
        // and calculating sum of product
        for (let i = 0; i <n; i++) {
            sop += (a[i] * b[i]);
        }

        return sop;
    }

    // Driver code
        let A = [ 1, 2, 3 ];
        let B = [ 4, 5, 1 ];

        document.write(maximumSOP(A, B));

// This code is contributed by Mayank Tyagi

</script>

Output

24

Complexity Analysis:

• Time Complexity: O(nlog(n))
• Auxiliary Space: O(1)