Minimize sum of smallest elements from K subsequences of length L

Given an array arr[] of size N, the task is to find the minimum possible sum by extracting the smallest element from any K subsequences from arr[] of length L such that each of the subsequences have no shared element. If it is not possible to get the required sum, print -1.

Examples:

Input: arr[] = {2, 15, 5, 1, 35, 16, 67, 10}, K = 3, L = 2 
Output: 8 
Explanation:
Three subsequences of length 2 can be {1, 35}, {2, 15}, {5, 16} 
Minimum element of {1, 35} is 1. 
Minimum element of {2, 15} is 2. 
Minimum element of {5, 16} is 5. 
Their Sum is equal to 8 which is the minimum possible.

Input: arr[] = {19, 11, 21, 16, 22, 18, 14, 12}, K = 3, L = 3 
Output: -1 
Explanation:
It is not possible to construct 3 subsequences of length 3 from arr[]. 

Approach:
To optimize the above approach, we need to observe the following details: 

• The K smallest elements of the array contribute to finding the minimum sum of the smallest elements of K subsequences.
• The length of the array must be greater than or equal to (K * L) in order to form K subsequences of length L.

Follow the steps below to solve the problem: 

• Check if the size of the array arr[] is greater than equal to (K * L).
• If so, sort the array arr[] and print the sum of the first K elements of the array after sorting.
• Otherwise, return -1.

Below is the implementation of the above approach:

C++

// C++ Program to find the minimum
// possible sum of the smallest
// elements from K subsequences

#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum sum
int findMinSum(int arr[], int K,
               int L, int size)
{

    if (K * L > size)
        return -1;

    int minsum = 0;

    // Sort the array
    sort(arr, arr + size);

    // Calculate sum of smallest
    // K elements
    for (int i = 0; i < K; i++)
        minsum += arr[i];

    // Return the sum
    return minsum;
}

// Driver Code
int main()
{
    int arr[] = { 2, 15, 5, 1,
                  35, 16, 67, 10 };
    int K = 3;
    int L = 2;

    int length = sizeof(arr)
                / sizeof(arr[0]);

    cout << findMinSum(arr, K,
                       L, length);

    return 0;
}

Java

// Java program to find the minimum
// possible sum of the smallest
// elements from K subsequences
import java.util.Arrays;

class GFG{

// Function to find the minimum sum
static int findMinSum(int []arr, int K,
                      int L, int size)
{
    if (K * L > size)
        return -1;

    int minsum = 0;

    // Sort the array
    Arrays.sort(arr);

    // Calculate sum of smallest
    // K elements
    for(int i = 0; i < K; i++)
        minsum += arr[i];

    // Return the sum
    return minsum;
}

// Driver Code
public static void main(String args[])
{
    int arr[] = { 2, 15, 5, 1,
                  35, 16, 67, 10 };
    int K = 3;
    int L = 2;
    int length = arr.length;

    System.out.print(findMinSum(arr, K,
                                L, length));
}
}

// This code is contributed by Ritik Bansal

Python3

# Python3 program to find the minimum
# possible sum of the smallest
# elements from K subsequences

# Function to find the minimum sum


def findMinSum(arr, K, L, size):

    if (K * L > size):
        return -1

    minsum = 0

    # Sort the array
    arr.sort()

    # Calculate sum of smallest
    # K elements
    for i in range(K):
        minsum += arr[i]

    # Return the sum
    return minsum


# Driver code
if __name__ == '__main__':

    arr = [2, 15, 5, 1,
           35, 16, 67, 10]
    K = 3
    L = 2

    length = len(arr)

    print(findMinSum(arr, K, L, length))

# This code is contributed by Shivam Singh

C#

// C# program to find the minimum
// possible sum of the smallest
// elements from K subsequences
using System;

class GFG{

// Function to find the minimum sum
static int findMinSum(int []arr, int K,
                      int L, int size)
{
    if (K * L > size)
        return -1;

    int minsum = 0;

    // Sort the array
    Array.Sort(arr);

    // Calculate sum of smallest
    // K elements
    for(int i = 0; i < K; i++)
        minsum += arr[i];

    // Return the sum
    return minsum;
}

// Driver Code
public static void Main()
{
    int[] arr = { 2, 15, 5, 1,
                  35, 16, 67, 10 };
    int K = 3;
    int L = 2;
    int length = arr.Length;

    Console.Write(findMinSum(arr, K,
                             L, length));
}
}

// This code is contributed by code_hunt

Javascript

<script>
// Javascript program to find the minimum
// possible sum of the smallest
// elements from K subsequences

// Function to find the minimum sum
function findMinSum(arr, K, L, size)
{
    if (K * L > size)
        return -1;

    let minsum = 0;

    // Sort the array
    arr.sort((a, b) => a - b);

    // Calculate sum of smallest
    // K elements
    for(let i = 0; i < K; i++)
        minsum += arr[i];

    // Return the sum
    return minsum;
}

    // Driver Code

    let arr = [ 2, 15, 5, 1,
                  35, 16, 67, 10 ];
    let K = 3;
    let L = 2;
    let length = arr.length;

   document.write(findMinSum(arr, K,
                                L, length));

</script>

Output

8

Time Complexity: O(N * log(N)) 
Space Complexity: O(1)