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

Difficulty Level : Hard
Last Updated : 17 Apr, 2020

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

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

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 

Output:

My Personal Notes

Related Articles

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

C++

Java

Python3

C#