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

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  • Difficulty Level : Hard
  • Last Updated : 04 Jun, 2022

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 chosen elements are stored in array B (B1, B2, B3…..Bk) 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 :
Let say, chosen 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

200

Time Complexity: O(n * log n)

Auxiliary Space: O(1)

Another Efficient Method: This problem can be solved efficiently using a Priority Queue. As we have to find the k smallest elements from the array in order to solve the expression so we use a priority queue which will find the k smallest elements in O(n*log(k)) time complexity where n is the size of the array and k is the number of smallest elements needed. Once we get the k smallest elements in the priority queue we will use the simplified expression given above to find the minimum value.

Below is the implementation of the above approach:  

C++




// C++ code to implement the approach
#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)
{
     
    // Using a Priority Queue
    // to find k smallest elements
    priority_queue<int> heap1;
  
    for (int i = 0; i < n; ++i) {
  
        // Insert elements into
        // the priority queue
        heap1.push(arr[i]);
  
        // If size of the priority
        // queue exceeds k then remove
        // the largest element from
        // the priority queue
        if (heap1.size() > k) {
            heap1.pop();
        }
    }
 
    int answer = 0;
 
    // Using first k elements from priority queue
    // to find the minimum value
    for (int i = 0; i < k; i++)
    {
        answer += heap1.top() * heap1.top();
        heap1.pop();
    }
 
    // 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;
}
 
// This code is contributed by Pushpesh Raj

Output

200

Time Complexity: O(k+nlog(k)) where n is the size of the array and k is the given number of elements to choose from.
Auxiliary Space: O(k) since the priority queue at any time holds k elements.


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