Maximum possible difference of two subsets of an array

Last Updated : 24 Mar, 2023

Given an array of n-integers. The array may contain repetitive elements but the highest frequency of any element must not exceed two. You have to make two subsets such that the difference of the sum of their elements is maximum and both of them jointly contain all elements of the given array along with the most important condition, no subset should contain repetitive elements.

Examples:

Input : arr[] = {5, 8, -1, 4}
Output : Maximum Difference = 18
Explanation : 
Let Subset A = {5, 8, 4} & Subset B = {-1}
Sum of elements of subset A = 17, of subset B = -1
Difference of Sum of Both subsets = 17 - (-1) = 18

Input : arr[] = {5, 8, 5, 4}
Output : Maximum Difference = 12
Explanation : 
Let Subset A = {5, 8, 4} & Subset B = {5}
Sum of elements of subset A = 17, of subset B = 5
Difference of Sum of Both subsets = 17 - 5 = 12

Before solving this question we have to take care of some given conditions, and they are listed as:

While building up the subsets, take care that no subset should contain repetitive elements. And for this, we can conclude that all such elements whose frequency are 2, going to be part of both subsets, and hence overall they don't have any impact on the difference of subset-sum. So, we can easily ignore them.
For making the difference of the sum of elements of both subset maximum we have to make subset in such a way that all positive elements belong to one subset and negative ones to other subsets.

Algorithm with time complexity O(n²):

for i=0 to n-1
    isSingleOccurrence = true;
    for  j= i+1 to n-1

        // if frequency of any element is two
        // make both equal to zero
        if arr[i] equals arr[j]
            arr[i] = arr[j] = 0
            isSingleOccurrence = false;
            break;
            
    if isSingleOccurrence == true
        if (arr[i] > 0)
            SubsetSum_1 += arr[i];
        else 
            SubsetSum_2 += arr[i];
return abs(SubsetSum_1 - SubsetSum2)

Implementation:

C++

// CPP find maximum difference of subset sum
#include <bits/stdc++.h>
using namespace std;

// function for maximum subset diff
int maxDiff(int arr[], int n)
{
    int SubsetSum_1 = 0, SubsetSum_2 = 0;
    for (int i = 0; i <= n - 1; i++) {

        bool isSingleOccurrence = true;
        for (int j = i + 1; j <= n - 1; j++) {

            // if frequency of any element is two
            // make both equal to zero
            if (arr[i] == arr[j]) {
                isSingleOccurrence = false;
                arr[i] = arr[j] = 0;
                break;
            }
        }
        if (isSingleOccurrence) {
            if (arr[i] > 0)
                SubsetSum_1 += arr[i];
            else
                SubsetSum_2 += arr[i];
        }
    }
    return abs(SubsetSum_1 - SubsetSum_2);
}

// driver program
int main()
{
    int arr[] = { 4, 2, -3, 3, -2, -2, 8 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << "Maximum Difference = " << maxDiff(arr, n);
    return 0;
}

Java

// java find maximum difference
// of subset sum
import java .io.*;

public class GFG {
    
    // function for maximum subset diff
    static int maxDiff(int []arr, int n)
    {
        int SubsetSum_1 = 0, SubsetSum_2 = 0;
        for (int i = 0; i <= n - 1; i++)
        {
            boolean isSingleOccurrence = true;
            for (int j = i + 1; j <= n - 1; j++)
            {
    
                // if frequency of any element
                // is two make both equal to
                // zero
                if (arr[i] == arr[j])
                {
                    isSingleOccurrence = false;
                    arr[i] = arr[j] = 0;
                    break;
                }
            }
            if (isSingleOccurrence)
            {
                if (arr[i] > 0)
                    SubsetSum_1 += arr[i];
                else
                    SubsetSum_2 += arr[i];
            }
        }
        
        return Math.abs(SubsetSum_1 - SubsetSum_2);
    }
    
    // driver program
    static public void main (String[] args)
    {
        int []arr = { 4, 2, -3, 3, -2, -2, 8 };
        int n = arr.length;
        
        System.out.println("Maximum Difference = "
                               + maxDiff(arr, n));
    }
}

// This code is contributed by vt_m.

Python3

# Python3 find maximum difference
# of subset sum

import math

# function for maximum subset diff
def maxDiff(arr, n) :
    SubsetSum_1 = 0
    SubsetSum_2 = 0
    for i in range(0, n) :

        isSingleOccurrence = True
        for j in range(i + 1, n) :

            # if frequency of any element
            # is two make both equal to 
            # zero
            if (arr[i] == arr[j]) : 
                isSingleOccurrence = False
                arr[i] = arr[j] = 0
                break

        if (isSingleOccurrence == True) :
            if (arr[i] > 0) :
                SubsetSum_1 += arr[i]
            else :
                SubsetSum_2 += arr[i]

    return abs(SubsetSum_1 - SubsetSum_2)

# Driver Code
arr = [4, 2, -3, 3, -2, -2, 8]
n = len(arr)
print ("Maximum Difference = {}"
               . format(maxDiff(arr, n)))

# This code is contributed by Manish Shaw
# (manishshaw1)

// C# find maximum difference of
// subset sum
using System;

public class GFG {
    
    // function for maximum subset diff
    static int maxDiff(int []arr, int n)
    {
        int SubsetSum_1 = 0, SubsetSum_2 = 0;
        for (int i = 0; i <= n - 1; i++)
        {
    
            bool isSingleOccurrence = true;
            for (int j = i + 1; j <= n - 1; j++)
            {
    
                // if frequency of any element
                // is two make both equal to
                // zero
                if (arr[i] == arr[j])
                {
                    isSingleOccurrence = false;
                    arr[i] = arr[j] = 0;
                    break;
                }
            }
            if (isSingleOccurrence)
            {
                if (arr[i] > 0)
                    SubsetSum_1 += arr[i];
                else
                    SubsetSum_2 += arr[i];
            }
        }
        
        return Math.Abs(SubsetSum_1 - SubsetSum_2);
    }
    
    // driver program
    static public void Main ()
    {
        int []arr = { 4, 2, -3, 3, -2, -2, 8 };
        int n = arr.Length;
        
        Console.WriteLine("Maximum Difference = "
                              + maxDiff(arr, n));
    }
}

// This code is contributed by vt_m.

PHP

<?php
// PHP find maximum difference
// of subset sum

// function for maximum subset diff
function maxDiff($arr, $n)
{
    $SubsetSum_1 = 0;
    $SubsetSum_2 = 0;
    for ($i = 0; $i <= $n - 1; $i++)
    {

        $isSingleOccurrence = true;
        for ($j = $i + 1; $j <= $n - 1; $j++)
        {

            // if frequency of any element is two
            // make both equal to zero
            if ($arr[$i] == $arr[$j]) 
            {
                $isSingleOccurrence = false;
                $arr[$i] = $arr[$j] = 0;
                break;
            }
        }
        if ($isSingleOccurrence) 
        {
            if ($arr[$i] > 0)
                $SubsetSum_1 += $arr[$i];
            else
                $SubsetSum_2 += $arr[$i];
        }
    }
    return abs($SubsetSum_1 - $SubsetSum_2);
}

    // Driver Code
    $arr = array(4, 2, -3, 3, -2, -2, 8);
    $n = sizeof($arr);
    echo "Maximum Difference = " , maxDiff($arr, $n);

// This code is contributed by nitin mittal
?>

JavaScript

<script>

// JavaScript Program to find maximum difference
// of subset sum

    // function for maximum subset diff
    function maxDiff(arr, n)
    {
        let SubsetSum_1 = 0, SubsetSum_2 = 0;
        for (let i = 0; i <= n - 1; i++)
        {
            let isSingleOccurrence = true;
            for (let j = i + 1; j <= n - 1; j++)
            {
     
                // if frequency of any element
                // is two make both equal to
                // zero
                if (arr[i] == arr[j])
                {
                    isSingleOccurrence = false;
                    arr[i] = arr[j] = 0;
                    break;
                }
            }
            if (isSingleOccurrence)
            {
                if (arr[i] > 0)
                    SubsetSum_1 += arr[i];
                else
                    SubsetSum_2 += arr[i];
            }
        }
         
        return Math.abs(SubsetSum_1 - SubsetSum_2);
    }
     

// Driver program

        let arr = [ 4, 2, -3, 3, -2, -2, 8 ];
        let n = arr.length;
         
        document.write("Maximum Difference = "
                               + maxDiff(arr, n));
        
        // This code is contributed by susmitakundugoaldanga.
</script>

Output

Maximum Difference = 20

Time Complexity O(n²)
Auxiliary Space: O(1)

Algorithm with time complexity O(n log n):

-> sort the array
-> for i =0 to n-2
      // consecutive two elements are not equal
      // add absolute arr[i] to result
      if arr[i] != arr[i+1]
          result += abs(arr[i])
      // else skip next element too
      else
          i++;
          
// special check for last two elements
-> if (arr[n-2] != arr[n-1])
    result += arr[n-1]

-> return result;

Implementation:

C++

// CPP find maximum difference of subset sum
#include <bits/stdc++.h>
using namespace std;

// function for maximum subset diff
int maxDiff(int arr[], int n)
{
    int result = 0;

    // sort the array
    sort(arr, arr + n);

    // calculate the result
    for (int i = 0; i < n - 1; i++) {
        if (arr[i] != arr[i + 1])
            result += abs(arr[i]);
        else
            i++;
    }

    // check for last element
    if (arr[n - 2] != arr[n - 1])
        result += abs(arr[n - 1]);

    // return result
    return result;
}

// driver program
int main()
{
    int arr[] = { 4, 2, -3, 3, -2, -2, 8 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << "Maximum Difference = " << maxDiff(arr, n);
    return 0;
}

Java

// java find maximum difference of
// subset sum
import java. io.*;
import java .util.*;

public class GFG {

    // function for maximum subset diff
    static int maxDiff(int []arr, int n)
    {
        int result = 0;
    
        // sort the array
        Arrays.sort(arr);
    
        // calculate the result
        for (int i = 0; i < n - 1; i++)
        {
            if (arr[i] != arr[i + 1])
                result += Math.abs(arr[i]);
            else
                i++;
        }
    
        // check for last element
        if (arr[n - 2] != arr[n - 1])
            result += Math.abs(arr[n - 1]);
    
        // return result
        return result;
    }
    
    // driver program
    static public void main (String[] args)
    {
        int[] arr = { 4, 2, -3, 3, -2, -2, 8 };
        int n = arr.length;
        
        System.out.println("Maximum Difference = "
                                + maxDiff(arr, n));
    }
}

// This code is contributed by vt_m.

Python 3

# Python 3 find maximum difference 
# of subset sum

# function for maximum subset diff
def maxDiff(arr, n):

    result = 0

    # sort the array
    arr.sort()

    # calculate the result
    for i in range(n - 1):
        if (abs(arr[i]) != abs(arr[i + 1])):
            result += abs(arr[i])

        else:
            pass

    # check for last element
    if (arr[n - 2] != arr[n - 1]):
        result += abs(arr[n - 1])

    # return result
    return result

# Driver Code
if __name__ == "__main__":
    
    arr = [ 4, 2, -3, 3, -2, -2, 8 ]
    n = len(arr)
    print("Maximum Difference = " ,
                   maxDiff(arr, n))

# This code is contributed by ita_c

// C# find maximum difference
// of subset sum
using System;

public class GFG {

    // function for maximum subset diff
    static int maxDiff(int []arr, int n)
    {
        int result = 0;
    
        // sort the array
        Array.Sort(arr);
    
        // calculate the result
        for (int i = 0; i < n - 1; i++)
        {
            if (arr[i] != arr[i + 1])
                result += Math.Abs(arr[i]);
            else
                i++;
        }
    
        // check for last element
        if (arr[n - 2] != arr[n - 1])
            result += Math.Abs(arr[n - 1]);
    
        // return result
        return result;
    }
    
    // driver program
    static public void Main ()
    {
        int[] arr = { 4, 2, -3, 3, -2, -2, 8 };
        int n = arr.Length;
        
        Console.WriteLine("Maximum Difference = "
                              + maxDiff(arr, n));
    }
}

// This code is contributed by vt_m.

PHP

<?php
// PHP find maximum difference of subset sum

// function for maximum subset diff
function maxDiff( $arr, $n)
{
    $result = 0;

    // sort the array
    sort($arr);

    // calculate the result
    for ( $i = 0; $i < $n - 1; $i++) 
    {
        if ($arr[$i] != $arr[$i + 1])
            $result += abs($arr[$i]);
        else
            $i++;
    }

    // check for last element
    if ($arr[$n - 2] != $arr[$n - 1])
        $result += abs($arr[$n - 1]);

    // return result
    return $result;
}

    // Driver Code
    $arr = array( 4, 2, -3, 3, -2, -2, 8 );
    $n = count($arr);
    echo "Maximum Difference = " 
        , maxDiff($arr, $n);
        
// This code is contributed by anuj_67.
?>

JavaScript

<script>

// Javascript find maximum difference of subset sum

// function for maximum subset diff
function maxDiff(arr, n)
{
    var result = 0;

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

    // calculate the result
    for (var i = 0; i < n - 1; i++) {
        if (arr[i] != arr[i + 1])
            result += Math.abs(arr[i]);
        else
            i++;
    }

    // check for last element
    if (arr[n - 2] != arr[n - 1])
        result += Math.abs(arr[n - 1]);

    // return result
    return result;
}

// driver program
var arr = [ 4, 2, -3, 3, -2, -2, 8 ];
var n = arr.length;
document.write( "Maximum Difference = " + maxDiff(arr, n));

</script>

Output

Maximum Difference = 20

Time Complexity: O(n log n)
Auxiliary Space: O(1)

Algorithm with time complexity O(n):

make hash table for positive elements:
    for all positive elements(arr[i])
        if frequency == 1
            SubsetSum_1 += arr[i];
make hash table for negative elements:
    for all negative elements
        if frequency == 1
            SubsetSum_2 += arr[i];
return abs(SubsetSum_1 - SubsetSum2)

Implementation:

C++

// CPP find maximum difference of subset sum
#include <bits/stdc++.h>
using namespace std;

// function for maximum subset diff
int maxDiff(int arr[], int n)
{
    unordered_map<int, int> hashPositive;
    unordered_map<int, int> hashNegative;

    int SubsetSum_1 = 0, SubsetSum_2 = 0;

    // construct hash for positive elements
    for (int i = 0; i <= n - 1; i++)
        if (arr[i] > 0)
            hashPositive[arr[i]]++;

    // calculate subset sum for positive elements
    for (int i = 0; i <= n - 1; i++)
        if (arr[i] > 0 && hashPositive[arr[i]] == 1)
            SubsetSum_1 += arr[i];

    // construct hash for negative elements
    for (int i = 0; i <= n - 1; i++)
        if (arr[i] < 0)
            hashNegative[abs(arr[i])]++;

    // calculate subset sum for negative elements
    for (int i = 0; i <= n - 1; i++)
        if (arr[i] < 0 && 
            hashNegative[abs(arr[i])] == 1)
            SubsetSum_2 += arr[i];

    return abs(SubsetSum_1 - SubsetSum_2);
}

// driver program
int main()
{
    int arr[] = { 4, 2, -3, 3, -2, -2, 8 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << "Maximum Difference = " << maxDiff(arr, n);
    return 0;
}

Java

// Java find maximum 
// difference of subset sum 
import java.util.*;
class GFG{
  
// Function for maximum subset diff 
public static int maxDiff(int arr[], 
                          int n) 
{ 
  HashMap<Integer, 
          Integer> hashPositive = new HashMap<>();
  HashMap<Integer, 
          Integer> hashNegative = new HashMap<>(); 

  int SubsetSum_1 = 0, 
      SubsetSum_2 = 0; 

  // Construct hash for 
  // positive elements 
  for (int i = 0; i <= n - 1; i++) 
  {
    if (arr[i] > 0) 
    {
      if(hashPositive.containsKey(arr[i]))
      {
        hashPositive.replace(arr[i], 
        hashPositive.get(arr[i]) + 1);
      }
      else
      {
        hashPositive.put(arr[i], 1);
      }
    }
  }

  // Calculate subset sum 
  // for positive elements 
  for (int i = 0; i <= n - 1; i++) 
  {
    if(arr[i] > 0 && 
       hashPositive.containsKey(arr[i]))
    {
      if(hashPositive.get(arr[i]) == 1)
      {
        SubsetSum_1 += arr[i];
      }
    }
  }

  // Construct hash for 
  // negative elements 
  for (int i = 0; i <= n - 1; i++) 
  {
    if (arr[i] < 0)
    {
      if(hashNegative.containsKey(Math.abs(arr[i])))
      {
        hashNegative.replace(Math.abs(arr[i]), 
        hashNegative.get(Math.abs(arr[i])) + 1);
      }
      else
      {
        hashNegative.put(Math.abs(arr[i]), 1);
      }
    }
  }

  // Calculate subset sum for
  // negative elements 
  for (int i = 0; i <= n - 1; i++) 
  {
    if (arr[i] < 0 && 
        hashNegative.containsKey(Math.abs(arr[i])))
    {
      if(hashNegative.get(Math.abs(arr[i])) == 1)
      {
        SubsetSum_2 += arr[i]; 
      }
    }
  }

  return Math.abs(SubsetSum_1 - SubsetSum_2); 
} 

// Driver code
public static void main(String[] args) 
{
  int arr[] = {4, 2, -3, 3, 
               -2, -2, 8}; 
  int n = arr.length;
  System.out.print("Maximum Difference = " + 
                    maxDiff(arr, n));
}
}

// This code is contributed by divyeshrabadiya07

Python3

# Python3 find maximum difference of subset sum

# function for maximum subset diff
def maxDiff(arr, n):

    hashPositive = dict()
    hashNegative = dict()

    SubsetSum_1, SubsetSum_2 = 0, 0

    # construct hash for positive elements
    for i in range(n):
        if (arr[i] > 0):
            hashPositive[arr[i]] = \
                hashPositive.get(arr[i], 0) + 1

    # calculate subset sum for positive elements
    for i in range(n):
        if (arr[i] > 0 and arr[i] in 
            hashPositive.keys() and 
            hashPositive[arr[i]] == 1):
            SubsetSum_1 += arr[i]

    # construct hash for negative elements
    for i in range(n):
        if (arr[i] < 0):
            hashNegative[abs(arr[i])] = \
                hashNegative.get(abs(arr[i]), 0) + 1

    # calculate subset sum for negative elements
    for i in range(n):
        if (arr[i] < 0 and abs(arr[i]) in 
            hashNegative.keys() and 
            hashNegative[abs(arr[i])] == 1):
            SubsetSum_2 += arr[i]

    return abs(SubsetSum_1 - SubsetSum_2)

# Driver Code
arr = [4, 2, -3, 3, -2, -2, 8]
n = len(arr)
print("Maximum Difference =", maxDiff(arr, n))

# This code is contributed by mohit kumar

// C# find maximum 
// difference of subset sum 
using System;
using System.Collections.Generic;

class GFG {

    // Function for maximum subset diff 
    static int maxDiff(int[] arr, int n) 
    { 
      Dictionary<int, int> hashPositive = 
        new Dictionary<int, int>();
      Dictionary<int, int> hashNegative = 
        new Dictionary<int, int>();
     
      int SubsetSum_1 = 0, SubsetSum_2 = 0; 
     
      // Construct hash for 
      // positive elements 
      for (int i = 0; i <= n - 1; i++) 
      {
        if (arr[i] > 0) 
        {
          if(hashPositive.ContainsKey(arr[i]))
          {
            hashPositive[arr[i]] += 1;
          }
          else
          {
            hashPositive.Add(arr[i], 1);
          }
        }
      }
     
      // Calculate subset sum 
      // for positive elements 
      for (int i = 0; i <= n - 1; i++) 
      {
        if(arr[i] > 0 && hashPositive.ContainsKey(arr[i]))
        {
          if(hashPositive[arr[i]] == 1)
          {
            SubsetSum_1 += arr[i];
          }
        }
      }
     
      // Construct hash for 
      // negative elements 
      for (int i = 0; i <= n - 1; i++) 
      {
        if (arr[i] < 0)
        {
          if(hashNegative.ContainsKey(Math.Abs(arr[i])))
          {
            hashNegative[(Math.Abs(arr[i]))] += 1; 
          }
          else
          {
            hashNegative.Add(Math.Abs(arr[i]), 1);
          }
        }
      }
     
      // Calculate subset sum for
      // negative elements 
      for (int i = 0; i <= n - 1; i++) 
      {
        if (arr[i] < 0 && 
            hashNegative.ContainsKey(Math.Abs(arr[i])))
        {
          if(hashNegative[(Math.Abs(arr[i]))] == 1)
          {
            SubsetSum_2 += arr[i]; 
          }
        }
      }
     
      return Math.Abs(SubsetSum_1 - SubsetSum_2); 
    }
  
  // Driver code  
  static void Main() {
      int[] arr = {4, 2, -3, 3, -2, -2, 8}; 
      int n = arr.Length;
      Console.WriteLine("Maximum Difference = " + 
                        maxDiff(arr, n));
  }
}

// This code is contributed by divesh072019

JavaScript

<script>
// Javascript find maximum
// difference of subset sum

// Function for maximum subset diff
function maxDiff(arr,n)
{
    let hashPositive = new Map();
  let hashNegative = new Map();
 
  let SubsetSum_1 = 0,
      SubsetSum_2 = 0;
 
  // Construct hash for
  // positive elements
  for (let i = 0; i <= n - 1; i++)
  {
    if (arr[i] > 0)
    {
      if(hashPositive.has(arr[i]))
      {
        hashPositive.set(arr[i],
        hashPositive.get(arr[i]) + 1);
      }
      else
      {
        hashPositive.set(arr[i], 1);
      }
    }
  }
 
  // Calculate subset sum
  // for positive elements
  for (let i = 0; i <= n - 1; i++)
  {
    if(arr[i] > 0 &&
       hashPositive.has(arr[i]))
    {
      if(hashPositive.get(arr[i]) == 1)
      {
        SubsetSum_1 += arr[i];
      }
    }
  }
 
  // Construct hash for
  // negative elements
  for (let i = 0; i <= n - 1; i++)
  {
    if (arr[i] < 0)
    {
      if(hashNegative.has(Math.abs(arr[i])))
      {
        hashNegative.set(Math.abs(arr[i]),
        hashNegative.get(Math.abs(arr[i])) + 1);
      }
      else
      {
        hashNegative.set(Math.abs(arr[i]), 1);
      }
    }
  }
 
  // Calculate subset sum for
  // negative elements
  for (let i = 0; i <= n - 1; i++)
  {
    if (arr[i] < 0 &&
        hashNegative.has(Math.abs(arr[i])))
    {
      if(hashNegative.get(Math.abs(arr[i])) == 1)
      {
        SubsetSum_2 += arr[i];
      }
    }
  }
 
  return Math.abs(SubsetSum_1 - SubsetSum_2);
        
}

// Driver code

let arr = [4, 2, -3, 3,
               -2, -2, 8];
let n = arr.length;
document.write("Maximum Difference = " +
                    maxDiff(arr, n));
                
    // This code is contributed by rag2127
</script>

Output

Maximum Difference = 20

Time Complexity: O(n)
Auxiliary Space: O(n)

Shivam.Pradhan

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