Minimize the sum of the squares of the sum of elements of each group the array is divided into

Given an array consisting of even number of elements, the task is to divide the array into M group of elements (every group must contain at least 2 elements) such that the sum of the squares of the sums of each group is minimized i.e.,
(sum_of_elements_of_group1)2 + (sum_of_elements_of_group2)2 + (sum_of_elements_of_group3)2 + (sum_of_elements_of_group4)2 + ….. + (sum_of_elements_of_groupM)2

Examples:

Input: arr[] = {5, 8, 13, 45, 6, 3}
Output: 2824
Groups can be (3, 45), (5, 13) and (6, 8)
(3 + 45)2 + (5 + 13)2 + (6 + 8)2 = 482 + 182 + 142 = 2304 + 324 + 196 = 2824

Input: arr[] = {53, 28, 143, 5}
Output: 28465

Approach: Our final sum depends on two factors:

  1. Sum of the elements of each group.
  2. The sum of squares of all such groups.

If we minimize both the factors mentioned above, we can minimize the result. To minimize the second factor we should make groups of minimum size i.e. just two elements. To minimize first factor we can pair smallest number with largest number, second smallest number to second largest number and so on.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the minimized sum
unsigned long long findAnswer(int n, 
                       vector<int>& arr)
{
  
    // Sort the array to pair the elements
    sort(arr.begin(), arr.end());
  
    // Variable to hold the answer
    unsigned long long sum = 0;
  
    // Pair smallest with largest, second
    // smallest with second largest, and 
    // so on
    for (int i = 0; i < n / 2; ++i) {
        sum += (arr[i] + arr[n - i - 1])
               * (arr[i] + arr[n - i - 1]);
    }
  
    return sum;
}
  
// Driver code
int main()
{
    std::vector<int> arr = { 53, 28, 143, 5 };
    int n = arr.size();
    cout << findAnswer(n, arr);
}

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Python3

# Python 3 implementation of the approach

# Function to return the minimized sum
def findAnswer(n, arr):

# Sort the array to pair the elements
arr.sort(reverse = False)

# Variable to hold the answer
sum = 0

# Pair smallest with largest, second
# smallest with second largest, and
# so on
for i in range(int(n / 2)):
sum += ((arr[i] + arr[n – i – 1]) *
(arr[i] + arr[n – i – 1]))

return sum

# Driver code
if __name__ == ‘__main__’:
arr = [53, 28, 143, 5]
n = len(arr)
print(findAnswer(n, arr))

# This code is contributed by
# Surendra_Gangwar

Output:

28465


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