Find minimum value of the expression by choosing K elements from given array

Difficulty Level : Hard
Last Updated : 16 Mar, 2021

Given an array of integers arr of size N, the task is to find the minimum possible of the expression by choosing exactly K(≤ N) integers form given array arr. Let say if choosen elements are stored in array B (B₁, B₂, B₃…..B_k) then value of expression:
$x = \sum_{i=1}^k\sum_{j=1}^k(B_i - B_j)^{2}$
Examples:

Input : arr[] = {2, 0, 9, 5}, k = 2
Output : 8
Let say, choosen elements are {2, 0}, then x = 8, which is minimum possible
Input : arr[] = {4, 21, 5, 3, 8}, k = 3
Output : 200

Approach :
The above expression can be simplified as:

$\sum_{i=1}^k\sum_{j=1}^k(B_i^2 + B_j^2 - 2*B_i*B_j)$
$\sum_{i=1}^k\sum_{j=1}^k{B_i^2} + \sum_{i=1}^k\sum_{j=1}^k{B_j^2} - 2\sum_{i=1}^k\sum_{j=1}^k{B_i*B_j}$
$k*\sum_{i=1}^k{B_i^2} + k*\sum_{i=1}^k{B_j^2} - 2\sum_{i=1}^k{B_i}*\sum_{j=1}^k{B_j}$
$2*k*\sum_{i=1}^k{B_i^2} - 2\sum_{i=1}^k{B_i^2}$
$(2*k-2)*\sum_{i=1}^k{B_i^2}$

So, all we need to do is select the k smallest elements from the array and solve the expression.
Below is the implementation of the above approach:

C++

// CPP program to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr
int minimumValue(int arr[], int n, int k)
{
 
    // Sorting the array for least k element selection
    sort(arr, arr + n);
 
    int answer = 0;
 
    // Select first k elements from sorted array
    for (int i = 0; i < k; i++)
        answer += arr[i] * arr[i];
 
    // Return value of solved expression
    return answer * (2 * k - 2);
}
 
// Driver code
int main()
{
    int arr[] = { 4, 21, 5, 3, 8 }, k = 3;
     
    int n = sizeof(arr) / sizeof(arr[0]);
     
    // Function call
    cout << minimumValue(arr, n, k);
     
    return 0;
}

Java

// JAVA program to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr 
import java.util.*;
 
class GFG{
 
// Function to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr
static int minimumValue(int arr[], int n, int k)
{
 
    // Sorting the array for least k element selection
    Arrays.sort(arr);
 
    int answer = 0;
 
    // Select first k elements from sorted array
    for (int i = 0; i < k; i++)
        answer += arr[i] * arr[i];
 
    // Return value of solved expression
    return answer * (2 * k - 2);
}
 
// Driver code
public static void main(String[] args)
{
    int arr[] = { 4, 21, 5, 3, 8 }, k = 3;
     
    int n = arr.length;
     
    // Function call
    System.out.print(minimumValue(arr, n, k));    
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python program to find the minimum
# possible of the expression by choosing
# exactly K(? N) integers form given array arr
 
# Function to find the minimum
# possible of the expression by
# choosing exactly K(? N) integers
# form given array arr
def minimumValue(arr, n, k):
 
    # Sorting the array for least k element selection
    arr.sort();
 
    answer = 0;
 
    # Select first k elements from sorted array
    for i in range(k):
        answer += arr[i] * arr[i];
 
    # Return value of solved expression
    return answer * (2 * k - 2);
 
# Driver code
if __name__ == '__main__':
    arr = [ 4, 21, 5, 3, 8 ];
    k = 3;
 
    n = len(arr);
 
    # Function call
    print(minimumValue(arr, n, k));
 
# This code is contributed by Rajput-Ji

C#

// C# program to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr 
using System;
 
class GFG{
  
// Function to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr
static int minimumValue(int []arr, int n, int k)
{
  
    // Sorting the array for least k element selection
    Array.Sort(arr);
  
    int answer = 0;
  
    // Select first k elements from sorted array
    for (int i = 0; i < k; i++)
        answer += arr[i] * arr[i];
  
    // Return value of solved expression
    return answer * (2 * k - 2);
}
  
// Driver code
public static void Main(String[] args)
{
    int []arr = { 4, 21, 5, 3, 8 };
    int k = 3;
      
    int n = arr.Length;
      
    // Function call
    Console.Write(minimumValue(arr, n, k));    
}
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
 
// JavaScript program to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr
 
// Function to find the minimum possible of the expression
// by choosing exactly K(? N) integers form given array arr
function minimumValue(arr, n, k)
{
 
    // Sorting the array for least k element selection
    arr.sort((a, b) => a - b);
 
    let answer = 0;
 
    // Select first k elements from sorted array
    for (let i = 0; i < k; i++)
        answer += arr[i] * arr[i];
 
    // Return value of solved expression
    return answer * (2 * k - 2);
}
 
// Driver code
 
    let arr = [ 4, 21, 5, 3, 8 ], k = 3;
     
    let n = arr.length;
     
    // Function call
    document.write(minimumValue(arr, n, k));
      
 
// This code is contributed by Surbhi Tyagi.
 
</script>

Output:

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Find minimum value of the expression by choosing K elements from given array

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