Partition into two subsets of lengths K and (N – k) such that the difference of sums is maximum

Given an array of non-negative integers of length N and an integer K. Partition the given array into two subsets of length K and N – K so that the difference between the sum of both subsets is maximum.

Examples :

Input : arr[] = {8, 4, 5, 2, 10}
        k = 2
Output : 17
Explanation :
Here, we can make first subset of length k = {4, 2}
and second subset of length N - k = {8, 5, 10}. Then,
the max_difference = (8 + 5 + 10) - (4 + 2) = 17.

Input : arr[] = {1, 1, 1, 1, 1, 1, 1, 1}
        k = 3
Output : 2
Explanation :
Here, subsets would be {1, 1, 1, 1, 1} and {1, 1, 1}.
So, max_difference would be 2

Choose k numbers with the largest possible sum. Then the solution obviously is k largest numbers. So that here greedy algorithm works – at each step we choose the largest possible number until we get all K numbers.

In this problem, we should divide the array of N numbers into two subsets of k and N – k numbers respectively. Consider two cases:

• The subset with the largest sum, among these two subsets, is a subset of K numbers. Then we want to maximize the sum in it since the sum in the second subset will only decrease if the sum in the first subarray will increase. So we are now in the sub-problem considered above and should choose k largest numbers.
• The subset with the largest sum, among these two subsets, is a subset of N – k numbers. Similar to the previous case, we then have to choose N – k largest numbers among all numbers.

Now, Let’s think about which of the two above cases actually gives the answer. We can easily see that a larger difference would be when more numbers are included in the group of largest numbers. Hence we could set M = max(k, N – k), find the sum of M largest numbers (let it be S1) and then the answer is S1 – (S – S1), where S is the sum of all numbers.

Below is the implementation of the above approach :  

C++

// C++ program to calculate max_difference between
// the sum of two subset of length k and N - k
#include <bits/stdc++.h>
using namespace std;

// Function to calculate max_difference
int maxDifference(int arr[], int N, int k)
{
    int M, S = 0, S1 = 0, max_difference = 0;

    // Sum of the array
    for (int i = 0; i < N; i++)
        S += arr[i];

    // Sort the array in descending order
    sort(arr, arr + N, greater<int>());
    M = max(k, N - k);
    for (int i = 0; i < M; i++)
        S1 += arr[i];

    // Calculating max_difference
    max_difference = S1 - (S - S1);
    return max_difference;
}

// Driver function
int main()
{
    int arr[] = { 8, 4, 5, 2, 10 };
    int N = sizeof(arr) / sizeof(arr[0]);
    int k = 2;
    cout << maxDifference(arr, N, k) << endl;
    return 0;
}

Java

// Java program to calculate max_difference between
// the sum of two subset of length k and N - k
import java.util.*;

class GFG
{

// Function to calculate max_difference
static int maxDifference(int arr[], int N, int k)
{
    int M, S = 0, S1 = 0, max_difference = 0;

    // Sum of the array
    for (int i = 0; i < N; i++)
        S += arr[i];
    int temp;

    // Sort the array in descending order
    for (int i = 0; i < N; i++)
    {
        for (int j = i + 1; j < N; j++)
        {
            if (arr[i] < arr[j])
            {
                temp = arr[i];
                arr[i] = arr[j];
                arr[j] = temp;
            }
        }
    }

    M = Math.max(k, N - k);
    for (int i = 0; i < M; i++)
        S1 += arr[i];

    // Calculating max_difference
    max_difference = S1 - (S - S1);
    return max_difference;
}

// Driver Code
public static void main(String args[])
{
    int arr[] = { 8, 4, 5, 2, 10 };
    int N = arr.length;
    int k = 2;
    System.out.println(maxDifference(arr, N, k));
}
}

// This code is contributed by
// Surendra_Gangwar

Python3

# Python3 code to calculate max_difference
# between the sum of two subset of
# length k and N - k

# Function to calculate max_difference
def maxDifference(arr, N, k ):
    S = 0
    S1 = 0
    max_difference = 0

    # Sum of the array
    for i in range(N):
        S += arr[i]

    # Sort the array in descending order
    arr.sort(reverse=True)
    M = max(k, N - k)
    for i in range( M):
        S1 += arr[i]

    # Calculating max_difference
    max_difference = S1 - (S - S1)
    return max_difference

# Driver Code
arr = [ 8, 4, 5, 2, 10 ]
N = len(arr)
k = 2
print(maxDifference(arr, N, k))

# This code is contributed by "Sharad_Bhardwaj".

C#

// C# program to calculate max_difference between
// the sum of two subset of length k and N - k
using System;

class GFG
{

// Function to calculate max_difference
static int maxDifference(int []arr, int N, int k)
{
    int M, S = 0, S1 = 0, max_difference = 0;

    // Sum of the array
    for (int i = 0; i < N; i++)
        S += arr[i];
    int temp;

    // Sort the array in descending order
    for (int i = 0; i < N; i++)
    {
        for (int j = i + 1; j < N; j++)
        {
            if (arr[i] < arr[j])
            {
                temp = arr[i];
                arr[i] = arr[j];
                arr[j] = temp;
            }
        }
    }

    M = Math.Max(k, N - k);
    for (int i = 0; i < M; i++)
        S1 += arr[i];

    // Calculating max_difference
    max_difference = S1 - (S - S1);
    return max_difference;
}

// Driver Code
public static void Main()
{
    int []arr = { 8, 4, 5, 2, 10 };
    int N = arr.Length;
    int k = 2;
    Console.Write(maxDifference(arr, N, k));
}
}

// This code is contributed by mohit kumar 29

PHP

<?php
// PHP program to calculate
// max_difference between
// the sum of two subset
// of length k and N - k

// Function to calculate
// max_difference
function maxDifference($arr, $N, $k)
{
    $M; $S = 0; $S1 = 0;
    $max_difference = 0;

    // Sum of the array
    for ($i = 0; $i < $N; $i++)
        $S += $arr[$i];

    // Sort the array in
    // descending order
    rsort($arr);
    $M = max($k, $N - $k);
    for ($i = 0; $i < $M; $i++)
        $S1 += $arr[$i];

    // Calculating
    // max_difference
    $max_difference = $S1 - ($S - $S1);
    return $max_difference;
}

// Driver Code
$arr = array(8, 4, 5, 2, 10);
$N = count($arr);
$k = 2;
echo maxDifference($arr, $N, $k);

// This code is contributed
// by anuj_67.
?>

Javascript

<script>

// Javascript program to calculate max_difference
// between the sum of two subset of length
// k and N - k

// Function to calculate max_difference
function maxDifference(arr, N, k)
{
    let M, S = 0, S1 = 0, max_difference = 0;

    // Sum of the array
    for(let i = 0; i < N; i++)
        S += arr[i];

    let temp;

    // Sort the array in descending order
    for(let i = 0; i < N; i++)
    {
        for(let j = i + 1; j < N; j++)
        {
            if (arr[i] < arr[j])
            {
                temp = arr[i];
                arr[i] = arr[j];
                arr[j] = temp;
            }
        }
    }

    M = Math.max(k, N - k);
    for(let i = 0; i < M; i++)
        S1 += arr[i];

    // Calculating max_difference
    max_difference = S1 - (S - S1);
    return max_difference;
}

// Driver code
let arr = [ 8, 4, 5, 2, 10 ];
let N = arr.length;
let k = 2;

document.write(maxDifference(arr, N, k));

// This code is contributed by divyeshrabadiya07

</script>

Output

17

Time Complexity: O(N log N), to sort the array 
Auxiliary Space: O(1), as no extra space is used

Further Optimizations : We can use Heap (or priority queue) to find M largest elements efficiently. Refer k largest(or smallest) elements in an array for details.