Maximum possible difference of two subsets of an array
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#
// 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> |
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#
// 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> |
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#
// 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> |
Maximum Difference = 20
Time Complexity: O(n)
Auxiliary Space: O(n)
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