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Maximize sum of second minimums of each K length partitions of the array
  • Last Updated : 01 Feb, 2021

Given an array A[] of size N and a positive integer K ( which will always be a factor of N), the task is to find the maximum possible sum of the second smallest elements of each partition of the array by partitioning the array into (N / K) partitions of equal size.

Examples: 

Input: A[] = {2, 3, 1, 4, 7, 5, 6, 1}, K = 4
Output: 7
Explanation: Split the array as {1, 2, 3, 4} and {1, 5, 6, 7}. Therefore, sum = 2 + 5 = 7, which is the maximum possible sum.

Input: A[] = {12, 43, 15, 32, 45, 23}, K = 3
Output : 66
Explanation: Split the array as {12, 23, 32} and {15, 43, 45}. Therefore, sum = 23 + 43 = 66, which is the maximum possible sum.

Approach: The idea is to sort the given array in ascending order and in order to maximize the required sum, divide the first N / K elements of A[] to each of the arrays as their first term, then choose every (K – 1)th element of A[] starting from N/K



Follow the steps below to solve the problem: 

  • Sort the array A[] in increasing order.
  • Initialize sum with 0 to store the required sum.
  • Now, initialize a variable i with N / K.
  • While i is less than N, perform the following steps:
    • Increment sum by A[i].
    • Increment i by K – 1.
  • After traversing, print sum as the required answer.

Below is the implementation of the above approach: 

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the maximum sum of
// second smallest of each partition
// of size K
void findSum(int A[], int N, int K)
{
 
    // Sort the array A[]
    // in ascending order
    sort(A, A + N);
 
    // Store the maximum sum of second
    // smallest of each partition
    // of size K
    int sum = 0;
 
    // Select every (K-1)th element as
    // second smallest element
    for (int i = N / K; i < N; i += K - 1) {
 
        // Update sum
        sum += A[i];
    }
 
    // Print the maximum sum
    cout << sum;
}
 
// Driver Code
int main()
{
 
    // Given size of partitions
    int K = 4;
 
    // Given array A[]
    int A[] = { 2, 3, 1, 4, 7, 5, 6, 1 };
 
    // Size of the given array
    int N = sizeof(A) / sizeof(A[0]);
 
    // Function Call
    findSum(A, N, K);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to find the maximum sum of
// second smallest of each partition
// of size K
static void findSum(int A[], int N, int K)
{
     
    // Sort the array A[]
    // in ascending order
    Arrays.sort(A);
     
    // Store the maximum sum of second
    // smallest of each partition
    // of size K
    int sum = 0;
 
    // Select every (K-1)th element as
    // second smallest element
    for(int i = N / K; i < N; i += K - 1)
    {
         
        // Update sum
        sum += A[i];
    }
 
    // Print the maximum sum
    System.out.print(sum);
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given size of partitions
    int K = 4;
 
    // Given array A[]
    int A[] = { 2, 3, 1, 4, 7, 5, 6, 1 };
 
    // Size of the given array
    int N = A.length;
 
    // Function Call
    findSum(A, N, K);
}
}
 
// This code is contributed by shikhasingrajput

Python3




# Python3 program for the above approach
 
# Function to find the maximum sum of
# second smallest of each partition
# of size K
def findSum(A, N, K):
   
    # Sort the array A
    # in ascending order
    A.sort();
 
    # Store the maximum sum of second
    # smallest of each partition
    # of size K
    sum = 0;
 
    # Select every (K-1)th element as
    # second smallest element
    for i in range(N // K, N, K - 1):
       
        # Update sum
        sum += A[i];
 
    # Prthe maximum sum
    print(sum);
 
# Driver Code
if __name__ == '__main__':
   
    # Given size of partitions
    K = 4;
 
    # Given array A
    A = [2, 3, 1, 4, 7, 5, 6, 1];
 
    # Size of the given array
    N = len(A);
 
    # Function Call
    findSum(A, N, K);
 
    # This code contributed by shikhasingrajput

C#




// C# program for the above approach
using System;
 
class GFG{
 
// Function to find the maximum sum of
// second smallest of each partition
// of size K
static void findSum(int []A, int N, int K)
{
     
    // Sort the array []A
    // in ascending order
    Array.Sort(A);
     
    // Store the maximum sum of second
    // smallest of each partition
    // of size K
    int sum = 0;
 
    // Select every (K-1)th element as
    // second smallest element
    for(int i = N / K; i < N; i += K - 1)
    {
         
        // Update sum
        sum += A[i];
    }
     
    // Print the maximum sum
    Console.Write(sum);
}
 
// Driver Code
public static void Main(String[] args)
{
     
    // Given size of partitions
    int K = 4;
 
    // Given array []A
    int []A = { 2, 3, 1, 4, 7, 5, 6, 1 };
 
    // Size of the given array
    int N = A.Length;
 
    // Function Call
    findSum(A, N, K);
}
}
 
// This code is contributed by shikhasingrajput
Output: 
7

 

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

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