Find maximum subset sum formed by partitioning any subset of array into 2 partitions with equal sum

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Given an array A containing N elements. Partition any subset of this array into two disjoint subsets such that both the subsets have an identical sum. Obtain the maximum sum that can be obtained after partitioning.

Note: It is not necessary to partition the entire array, that is any element might not contribute to any of the partition.

Examples:

Input: A = [1, 2, 3, 6]
Output: 6
Explanation: We have two disjoint subsets {1, 2, 3} and {6}, which have the same sum = 6

Input: A = [1, 2, 3, 4, 5, 6]
Output: 10
Explanation: We have two disjoint subsets {2, 3, 5} and {4, 6}, which have the same sum = 10.

Input: A = [1, 2]
Output: 0
Explanation: No subset can be partitioned into 2 disjoint subsets with identical sum

Naive Approach:
The above problem can be solved by brute force method using recursion. All the elements have three possibilities. Either it will contribute to partition 1 or partition 2 or will not be included in any of the partitions. We will perform these three operations on each of the elements and proceed to the next element in each recursive step.

Below is the implementation of the above approach:

C++

// CPP implementation for the
// above mentioned recursive approach
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to find the maximum subset sum
int maxSum(int p0, int p1, int a[], int pos, int n)
{
    if (pos == n) {
        if (p0 == p1)
            return p0;
        else
            return 0;
    }
    // Ignore the current element
    int ans = maxSum(p0, p1, a, pos + 1, n);
 
    // including element in partition 1
    ans = max(ans, maxSum(p0 + a[pos], p1, a, pos + 1, n));
 
    // including element in partition 2
    ans = max(ans, maxSum(p0, p1 + a[pos], a, pos + 1, n));
    return ans;
}
 
// Driver code
int main()
{
    // size of the array
    int n = 4;
    int a[n] = { 1, 2, 3, 6 };
    cout << maxSum(0, 0, a, 0, n);
    return 0;
}

Java

// Java implementation for the
// above mentioned recursive approach
class GFG {
     
    // Function to find the maximum subset sum
    static int maxSum(int p0, int p1, int a[], int pos, int n)
    {
        if (pos == n) {
            if (p0 == p1)
                return p0;
            else
                return 0;
        }
 
        // Ignore the current element
        int ans = maxSum(p0, p1, a, pos + 1, n);
     
        // including element in partition 1
        ans = Math.max(ans, maxSum(p0 + a[pos], p1, a, pos + 1, n));
     
        // including element in partition 2
        ans = Math.max(ans, maxSum(p0, p1 + a[pos], a, pos + 1, n));
        return ans;
    }
     
    // Driver code
    public static void main (String[] args)
    {
        // size of the array
        int n = 4;
        int a[] = { 1, 2, 3, 6 };
        System.out.println(maxSum(0, 0, a, 0, n));
         
    }
}
 
// This code is contributed by AnkitRai01

Python3

# Python3 implementation for the
# above mentioned recursive approach
 
# Function to find the maximum subset sum
def maxSum(p0, p1, a, pos, n) :
 
    if (pos == n) :
        if (p0 == p1) :
            return p0;
        else :
            return 0;
     
    # Ignore the current element
    ans = maxSum(p0, p1, a, pos + 1, n);
 
    # including element in partition 1
    ans = max(ans, maxSum(p0 + a[pos], p1, a, pos + 1, n));
 
    # including element in partition 2
    ans = max(ans, maxSum(p0, p1 + a[pos], a, pos + 1, n));
     
    return ans;
 
# Driver code
if __name__ == "__main__" :
 
    # size of the array
    n = 4;
    a = [ 1, 2, 3, 6 ];
     
    print(maxSum(0, 0, a, 0, n));
 
# This code is contributed by AnkitRai01

C#

// C# implementation for the
// above mentioned recursive approach
 
using System;
 
public class GFG {
     
    // Function to find the maximum subset sum
    static int maxSum(int p0, int p1, int []a, int pos, int n)
    {
        if (pos == n) {
            if (p0 == p1)
                return p0;
            else
                return 0;
        }
 
        // Ignore the current element
        int ans = maxSum(p0, p1, a, pos + 1, n);
     
        // including element in partition 1
        ans = Math.Max(ans, maxSum(p0 + a[pos], p1, a, pos + 1, n));
     
        // including element in partition 2
        ans = Math.Max(ans, maxSum(p0, p1 + a[pos], a, pos + 1, n));
        return ans;
    }
     
    // Driver code
    public static void Main (string[] args)
    {
        // size of the array
        int n = 4;
        int []a = { 1, 2, 3, 6 };
        Console.WriteLine(maxSum(0, 0, a, 0, n));
         
    }
}
 
// This code is contributed by AnkitRai01

Javascript

<script>
 
// Javascript implementation for the
// above mentioned recursive approach
 
// Function to find the maximum subset sum
function maxSum(p0, p1, a, pos, n)
{
    if (pos == n)
    {
        if (p0 == p1)
            return p0;
        else
            return 0;
    }
     
    // Ignore the current element
    var ans = maxSum(p0, p1, a, pos + 1, n);
 
    // Including element in partition 1
    ans = Math.max(ans, maxSum(
        p0 + a[pos], p1, a, pos + 1, n));
 
    // Including element in partition 2
    ans = Math.max(ans, maxSum(
        p0, p1 + a[pos], a, pos + 1, n));
    return ans;
}
 
// Driver code
 
// Size of the array
var n = 4;
var a = [ 1, 2, 3, 6 ];
 
document.write(maxSum(0, 0, a, 0, n));
 
// This code is contributed by importantly
 
</script>

Output

Time Complexity: $O(3^n)$
Auxiliary Space: O(n)

Memoization: Aa we can see there are multiple overlapping subproblems so instead of solving them again and again we can store each recursive call result in an array and use it .

C++

// CPP implementation for the
// above mentioned recursive approach
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to find the maximum subset sum
int maxSum(int p0, int p1, int a[], int pos, int n,
           vector<vector<int> >& dp)
{
    if (pos == n) {
        if (p0 == p1)
            return p0;
        else
            return 0;
    }
   //if the value is already computed then return that previous computed value.
    if (dp[pos][p0] != -1) {
        return dp[pos][p0];
    }
   // Ignore the current element
    int ans = maxSum(p0, p1, a, pos + 1, n, dp);
 
    // including element in partition 1
    ans = max(ans,
              maxSum(p0 + a[pos], p1, a, pos + 1, n, dp));
 
    // including element in partition 2
    ans = max(ans,
              maxSum(p0, p1 + a[pos], a, pos + 1, n, dp));
    return dp[pos][p0] = ans;
}
 
int maxSum(int p0, int p1, int a[], int pos, int n)
{
    int sum = 0;
    for (int i = 0; i < n; i++) {
        sum += a[i];
    }
    vector<vector<int> > dp(n, vector<int>(sum + 1, -1));
    return maxSum(p0, p1, a, pos, n, dp);
}
// Driver code
int main()
{
    // size of the array
    int n = 4;
    int a[n] = { 1, 2, 3, 6 };
    cout << maxSum(0, 0, a, 0, n);
 
    return 0;
}

Java

// Java implementation for the
// above mentioned recursive approach
import java.util.Arrays;
class GFG {
 
    // Function to find the maximum subset sum
    static int maxSum(int p0, int p1, int a[], int pos,
                      int n, int[][] dp)
    {
        if (pos == n) {
            if (p0 == p1)
                return p0;
            else
                return 0;
        }
        //if the value is already computed then return that previous computed value.
        if (dp[pos][p0] != -1) {
            return dp[pos][p0];
        }
 
        // Ignore the current element
 
        int ans = maxSum(p0, p1, a, pos + 1, n, dp);
 
        // including element in partition 1
        ans = Math.max(
            ans, maxSum(p0 + a[pos], p1, a, pos + 1, n,dp));
 
        // including element in partition 2
        ans = Math.max(
            ans, maxSum(p0, p1 + a[pos], a, pos + 1, n,dp));
        return dp[pos][p0] = ans;
    }
    static int maxSum(int p0, int p1, int a[], int pos,
                      int n)
    {
        int sum = 0;
        for (int i = 0; i < n; i++) {
            sum += a[i];
        }
        int dp[][] = new int[n][sum + 1];
        for (int row[] : dp) {
            Arrays.fill(row, -1);
        }
        return maxSum(p0, p1, a, pos, n, dp);
    }
 
    // Driver code
    public static void main(String[] args)
    {
        // size of the array
        int n = 4;
        int a[] = { 1, 2, 3, 6 };
        System.out.println(maxSum(0, 0, a, 0, n));
    }
}

Python3

# Python code for the above approach
def maxSum(p0, p1, a, pos, n, dp):
   
    # Base case : If we have reached the end of the array
    if pos == n:
       
        # If both partitions have equal sum, return that sum
        if p0 == p1:
            return p0
           
        # If both partitions have different sum, return 0
        else:
            return 0
    # If the value is already computed, return that previous computed value
    if dp[pos][p0] != -1:
        return dp[pos][p0]
       
    # Ignore the current element
    ans = maxSum(p0, p1, a, pos + 1, n, dp)
 
    # including element in partition 1
    ans = max(ans, maxSum(p0 + a[pos], p1, a, pos + 1, n, dp))
 
    # including element in partition 2
    ans = max(ans, maxSum(p0, p1 + a[pos], a, pos + 1, n, dp))
    dp[pos][p0] = ans
    return dp[pos][p0]
 
def maxSumWrapper(a,n):
    sum = 0
    for i in range(n):
        sum += a[i]
    dp = [[-1 for i in range(sum+1)] for j in range(n)]
    return maxSum(0, 0, a, 0, n, dp)
 
# Driver code
if __name__ == '__main__':
   
    # size of the array
    n = 4
    a = [1, 2, 3, 6]
    print(maxSumWrapper(a, n))
 
    # This code is contributed by pradeepkumarppk2003

C#

// C# implementation for the
// above mentioned recursive approach
using System;
 
class GFG
{
 
    // Function to find the maximum subset sum
    static int maxSum(int p0, int p1, int[] a, int pos,
                      int n, int[,] dp)
    {
        if (pos == n)
        {
            if (p0 == p1)
                return p0;
            else
                return 0;
        }
        //if the value is already computed then return 
      // that previous computed value.
        if (dp[pos, p0] != -1)
        {
            return dp[pos, p0];
        }
 
        // Ignore the current element
 
        int ans = maxSum(p0, p1, a, pos + 1, n, dp);
 
        // including element in partition 1
        ans = Math.Max(
            ans, maxSum(p0 + a[pos], p1, a, pos + 1, n, dp));
 
        // including element in partition 2
        ans = Math.Max(
            ans, maxSum(p0, p1 + a[pos], a, pos + 1, n, dp));
        return dp[pos, p0] = ans;
    }
    static int maxSum(int p0, int p1, int[] a, int pos,
                      int n)
    {
        int sum = 0;
        for (int i = 0; i < n; i++)
        {
            sum += a[i];
        }
        int[,] dp = new int[n, sum + 1];
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < sum + 1; j++)
            {
                dp[i, j] = -1;
            }
        }
        return maxSum(p0, p1, a, pos, n, dp);
    }
 
    // Driver code
    public static void Main(string[] args)
    {
        // size of the array
        int n = 4;
        int[] a = { 1, 2, 3, 6 };
        Console.WriteLine(maxSum(0, 0, a, 0, n));
    }
}
//This code is contributed by ik_9

Javascript

// Javascript implementation for the
// above mentioned recursive approach
 
 
// Function to find the maximum subset sum
function maxSum(p0, p1, a, pos, n,
    dp) {
    if (pos == n) {
        if (p0 == p1)
            return p0;
        else
            return 0;
    }
    //if the value is already computed then return that previous computed value.
    if (dp[pos][p0] != -1) {
        return dp[pos][p0];
    }
    // Ignore the current element
    let ans = maxSum(p0, p1, a, pos + 1, n, dp);
 
    // including element in partition 1
    ans = Math.max(ans,
        maxSum(p0 + a[pos], p1, a, pos + 1, n, dp));
 
    // including element in partition 2
    ans = Math.max(ans,
        maxSum(p0, p1 + a[pos], a, pos + 1, n, dp));
    return dp[pos][p0] = ans;
}
 
function maxsum(p0, p1, a, pos, n) {
    let sum = 0;
    for (let i = 0; i < n; i++) {
        sum += a[i];
    }
    let dp=new Array(n);
    for(let i=0;i<n;i++)
    {
        dp[i]=new Array(sum+1);
        for(let j=0;j<sum+1;j++)
        dp[i][j]=-1;
    }
    return maxSum(p0, p1, a, pos, n, dp);
}
// Driver code
 
// size of the array
let n = 4;
let a = [1, 2, 3, 6];
console.log(maxsum(0, 0, a, 0, n));

Output

Time Complexity: O(N*Sum), where Sum represents sum of all array elements.
Auxiliary Space: O(N*Sum) + O(N) .

Efficient Approach:
The above method can be optimized using Dynamic Programming method.
We will define our DP state as follows :

dp[i][j] = Max sum of group g0 considering the first i elements such that,
the difference between the sum of g0 and g1 is (sum of all elements – j), where j is the difference.
So, the answer would be dp[n][sum]

Now we might encounter, the difference between the sums is negative, lying in the range [-sum, +sum], where the sum is a summation of all elements. The minimum and maximum ranges occurring when one of the subsets is empty and the other one has all the elements. Due to this, in the DP state, we have defined j as (sum – diff). Thus, j will range from [0, 2*sum].

Below is the implementation of the above approach:

C++

// CPP implementation for the above mentioned
// Dynamic Programming  approach
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to find the maximum subset sum
int maxSum(int a[], int n)
{
    // sum of all elements
    int sum = 0;
    for (int i = 0; i < n; i++)
        sum += a[i];
 
    int limit = 2 * sum + 1;
 
    // bottom up lookup table;
    int dp[n + 1][limit];
 
    // initialising dp table with INT_MIN
    // where, INT_MIN means no solution
    for (int i = 0; i < n + 1; i++) {
        for (int j = 0; j < limit; j++)
            dp[i][j] = INT_MIN;
    }
 
    // Case when diff is 0
    dp[0][sum] = 0;
    for (int i = 1; i <= n; i++) {
        for (int j = 0; j < limit; j++) {
 
            // Putting ith element in g0
            if ((j - a[i - 1]) >= 0 && dp[i - 1][j - a[i - 1]] != INT_MIN)
 
                dp[i][j] = max(dp[i][j], dp[i - 1][j - a[i - 1]]
                                             + a[i - 1]);
 
            // Putting ith element in g1
            if ((j + a[i - 1]) < limit && dp[i - 1][j + a[i - 1]] != INT_MIN)
 
                dp[i][j] = max(dp[i][j], dp[i - 1][j + a[i - 1]]);
 
            // Ignoring ith element
            if (dp[i - 1][j] != INT_MIN)
 
                dp[i][j] = max(dp[i][j], dp[i - 1][j]);
        }
    }
 
    return dp[n][sum];
}
 
// Driver code
 
int main()
{
    int n = 4;
    int a[n] = { 1, 2, 3, 6 };
    cout << maxSum(a, n);
    return 0;
}

Java

// Java implementation for the above mentioned 
// Dynamic Programming approach 
class GFG {
     
    final static int INT_MIN = Integer.MIN_VALUE;
     
    // Function to find the maximum subset sum 
    static int maxSum(int a[], int n) 
    { 
        // sum of all elements 
        int sum = 0; 
        for (int i = 0; i < n; i++) 
            sum += a[i]; 
     
        int limit = 2 * sum + 1; 
     
        // bottom up lookup table; 
        int dp[][] = new int[n + 1][limit]; 
     
        // initialising dp table with INT_MIN 
        // where, INT_MIN means no solution 
        for (int i = 0; i < n + 1; i++) { 
            for (int j = 0; j < limit; j++) 
                dp[i][j] = INT_MIN; 
        } 
     
        // Case when diff is 0 
        dp[0][sum] = 0; 
        for (int i = 1; i <= n; i++) { 
            for (int j = 0; j < limit; j++) { 
     
                // Putting ith element in g0 
                if ((j - a[i - 1]) >= 0 && dp[i - 1][j - a[i - 1]] != INT_MIN) 
     
                    dp[i][j] = Math.max(dp[i][j], dp[i - 1][j - a[i - 1]] 
                                                + a[i - 1]); 
     
                // Putting ith element in g1 
                if ((j + a[i - 1]) < limit && dp[i - 1][j + a[i - 1]] != INT_MIN) 
     
                    dp[i][j] = Math.max(dp[i][j], dp[i - 1][j + a[i - 1]]); 
     
                // Ignoring ith element 
                if (dp[i - 1][j] != INT_MIN) 
     
                    dp[i][j] = Math.max(dp[i][j], dp[i - 1][j]); 
            } 
        } 
     
        return dp[n][sum]; 
    } 
     
    // Driver code 
    public static void main (String[] args) 
    { 
        int n = 4; 
        int []a = { 1, 2, 3, 6 }; 
        System.out.println(maxSum(a, n)); 
    } 
}
 
// This code is contributed by AnkitRai01

Python3

# Python3 implementation for the above mentioned 
# Dynamic Programming approach 
import numpy as np
import sys
 
INT_MIN = -(sys.maxsize - 1)
 
# Function to find the maximum subset sum 
def maxSum(a, n) : 
 
    # sum of all elements 
    sum = 0; 
    for i in range(n) :
        sum += a[i]; 
 
    limit = 2 * sum + 1; 
 
    # bottom up lookup table; 
    dp = np.zeros((n + 1,limit)); 
 
    # initialising dp table with INT_MIN 
    # where, INT_MIN means no solution 
    for i in range(n + 1) :
        for j in range(limit) :
            dp[i][j] = INT_MIN; 
 
    # Case when diff is 0 
    dp[0][sum] = 0; 
    for i in range(1, n + 1) :
        for j in range(limit) :
 
            # Putting ith element in g0 
            if ((j - a[i - 1]) >= 0 and dp[i - 1][j - a[i - 1]] != INT_MIN) :
 
                dp[i][j] = max(dp[i][j], dp[i - 1][j - a[i - 1]] 
                                            + a[i - 1]); 
 
            # Putting ith element in g1 
            if ((j + a[i - 1]) < limit and dp[i - 1][j + a[i - 1]] != INT_MIN) :
 
                dp[i][j] = max(dp[i][j], dp[i - 1][j + a[i - 1]]); 
 
            # Ignoring ith element 
            if (dp[i - 1][j] != INT_MIN) :
 
                dp[i][j] = max(dp[i][j], dp[i - 1][j]); 
                 
    return dp[n][sum]; 
 
# Driver code 
 
if __name__ == "__main__" : 
 
    n = 4; 
    a = [ 1, 2, 3, 6 ]; 
    print(maxSum(a, n)); 
 
# This code is contributed by Yash_R

C#

// C# implementation for the above mentioned 
// Dynamic Programming approach 
using System;
 
class GFG {
     
    static int INT_MIN = int.MinValue;
     
    // Function to find the maximum subset sum 
    static int maxSum(int []a, int n) 
    { 
        // sum of all elements 
        int sum = 0; 
        for (int i = 0; i < n; i++) 
            sum += a[i]; 
     
        int limit = 2 * sum + 1; 
     
        // bottom up lookup table; 
        int [,]dp = new int[n + 1,limit]; 
     
        // initialising dp table with INT_MIN 
        // where, INT_MIN means no solution 
        for (int i = 0; i < n + 1; i++) { 
            for (int j = 0; j < limit; j++) 
                dp[i,j] = INT_MIN; 
        } 
     
        // Case when diff is 0 
        dp[0,sum] = 0; 
        for (int i = 1; i <= n; i++) { 
            for (int j = 0; j < limit; j++) { 
     
                // Putting ith element in g0 
                if ((j - a[i - 1]) >= 0 && dp[i - 1,j - a[i - 1]] != INT_MIN) 
     
                    dp[i,j] = Math.Max(dp[i,j], dp[i - 1,j - a[i - 1]] 
                                                + a[i - 1]); 
     
                // Putting ith element in g1 
                if ((j + a[i - 1]) < limit && dp[i - 1,j + a[i - 1]] != INT_MIN) 
     
                    dp[i,j] = Math.Max(dp[i,j], dp[i - 1,j + a[i - 1]]); 
     
                // Ignoring ith element 
                if (dp[i - 1,j] != INT_MIN) 
     
                    dp[i,j] = Math.Max(dp[i,j], dp[i - 1,j]); 
            } 
        } 
     
        return dp[n,sum]; 
    } 
     
    // Driver code 
    public static void Main() 
    { 
        int n = 4; 
        int []a = { 1, 2, 3, 6 }; 
        Console.WriteLine(maxSum(a, n)); 
    } 
}
 
// This code is contributed by Yash_R

Javascript

<script>
 
 
// JavaScript implementation for the above mentioned
// Dynamic Programming  approach
 
// Function to find the maximum subset sum
function maxSum(a, n)
{
    // sum of all elements
    var sum = 0;
    for (var i = 0; i < n; i++)
        sum += a[i];
 
    var limit = 2 * sum + 1;
 
    // bottom up lookup table;
    var dp = Array.from(Array(n+1), ()=>Array(limit));
 
    // initialising dp table with -1000000000
    // where, -1000000000 means no solution
    for (var i = 0; i < n + 1; i++) {
        for (var j = 0; j < limit; j++)
            dp[i][j] = -1000000000;
    }
 
    // Case when diff is 0
    dp[0][sum] = 0;
    for (var i = 1; i <= n; i++) {
        for (var j = 0; j < limit; j++) {
 
            // Putting ith element in g0
            if ((j - a[i - 1]) >= 0 && 
            dp[i - 1][j - a[i - 1]] != -1000000000)
 
                dp[i][j] = Math.max(dp[i][j], 
                dp[i - 1][j - a[i - 1]] + a[i - 1]);
 
            // Putting ith element in g1
            if ((j + a[i - 1]) < limit && 
            dp[i - 1][j + a[i - 1]] != -1000000000)
 
                dp[i][j] = Math.max(dp[i][j], 
                dp[i - 1][j + a[i - 1]]);
 
            // Ignoring ith element
            if (dp[i - 1][j] != -1000000000)
 
                dp[i][j] = Math.max(dp[i][j], dp[i - 1][j]);
        }
    }
 
    return dp[n][sum];
}
 
// Driver code
var n = 4;
var a = [1, 2, 3, 6];
document.write( maxSum(a, n));
 
</script>

Output

Time Complexity: $O(N*Sum)$ , where Sum represents sum of all array elements.
Auxiliary Space: $O(N*Sum)$

Efficient Approach : using array instead of 2d matrix to optimize space complexity

In previous code we can se that dp[i][j] is dependent upon dp[i-1] or dp[i] so we can assume that dp[i-1] is previous row and dp[i] is current row.

Implementations Steps :

Create two vectors prev and curr each of size limit+1, where limit is a 2 * sum + 1.
Initialize them with base cases.
Now In previous code change dp[i] to curr and change dp[i-1] to prev to keep track only of the two main rows.
After every iteration update previous row to current row to iterate further.

Implementation :

C++

// CPP implementation for the above mentioned
// Dynamic Programming approach
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to find the maximum subset sum
int maxSum(int a[], int n)
{
    // sum of all elements
    int sum = 0;
    for (int i = 0; i < n; i++)
        sum += a[i];
 
    int limit = 2 * sum + 1;
 
 
    // initialising curr and prev vectors table with INT_MIN
    // where, INT_MIN means no solution
    vector<int>prev(limit +1 , INT_MIN);
    vector<int>curr(limit +1 , INT_MIN);
     
     
 
 
    // Case when diff is 0
    prev[sum] = 0;
     
    for (int i = 1; i <= n; i++) {
        for (int j = 0; j < limit; j++) {
 
            // Putting ith element in g0
            if ((j - a[i - 1]) >= 0 && prev[j - a[i - 1]] != INT_MIN)
 
                curr[j] = max(curr[j], prev[j - a[i - 1]]
                                            + a[i - 1]);
 
            // Putting ith element in g1
            if ((j + a[i - 1]) < limit && prev[j + a[i - 1]] != INT_MIN)
 
                curr[j] = max(curr[j], prev[j + a[i - 1]]);
 
            // Ignoring ith element
            if (prev[j] != INT_MIN)
 
                curr[j] = max(curr[j], prev[j]);
        }
        // assigning values of curr to prev vector to iterate further
        prev = curr;
    }
     
    // return answer
    return curr[sum];
}
 
// Driver code
int main()
{
    int n = 4;
    int a[n] = { 1, 2, 3, 6 };
     
    // function call
    cout << maxSum(a, n);
    return 0;
}

Java

// Java implementation for the above approach
// Dynamic Programming approach
 
import java.util.*;
 
public class Main 
{
 
  // Function to find the maximum subset sum
  public static int maxSum(int[] a, int n)
  {
 
    // sum of all elements
    int sum = 0;
    for (int i = 0; i < n; i++)
      sum += a[i];
 
    int limit = 2 * sum + 1;
 
    // initialising curr and prev vectors table with
    // INT_MIN where, INT_MIN means no solution
    int[] prev = new int[limit + 1];
    int[] curr = new int[limit + 1];
 
    Arrays.fill(prev, Integer.MIN_VALUE);
    Arrays.fill(curr, Integer.MIN_VALUE);
 
    // Case when diff is 0
    prev[sum] = 0;
 
    for (int i = 1; i <= n; i++) {
      for (int j = 0; j < limit; j++) {
 
        // Putting ith element in g0
        if ((j - a[i - 1]) >= 0
            && prev[j - a[i - 1]]
            != Integer.MIN_VALUE)
          curr[j] = Math.max(curr[j],
                             prev[j - a[i - 1]]
                             + a[i - 1]);
 
        // Putting ith element in g1
        if ((j + a[i - 1]) < limit
            && prev[j + a[i - 1]]
            != Integer.MIN_VALUE)
          curr[j] = Math.max(curr[j],
                             prev[j + a[i - 1]]);
 
        // Ignoring ith element
        if (prev[j] != Integer.MIN_VALUE)
          curr[j] = Math.max(curr[j], prev[j]);
      }
      // assigning values of curr to prev vector to
      // iterate further
      prev = curr.clone();
    }
 
    // return answer
    return curr[sum];
  }
 
  // Driver code
  public static void main(String[] args)
  {
    int n = 4;
    int[] a = { 1, 2, 3, 6 };
 
    // function call
    System.out.println(maxSum(a, n));
  }
}
 
// This code is contributed by sarojmcy2e

Python

def max_sum(a, n):
    # Sum of all elements
    total_sum = sum(a)
 
    # Calculate the limit for the array
    limit = 2 * total_sum + 1
 
    # Initialize curr and prev lists with float('-inf')
    prev = [float('-inf')] * (limit + 1)
    curr = [float('-inf')] * (limit + 1)
 
    # Case when diff is 0
    prev[total_sum] = 0
 
    for i in range(1, n + 1):
        for j in range(limit):
            # Putting the ith element in group 0
            if (j - a[i - 1]) >= 0 and prev[j - a[i - 1]] != float('-inf'):
                curr[j] = max(curr[j], prev[j - a[i - 1]] + a[i - 1])
 
            # Putting the ith element in group 1
            if (j + a[i - 1]) < limit and prev[j + a[i - 1]] != float('-inf'):
                curr[j] = max(curr[j], prev[j + a[i - 1]])
 
            # Ignoring the ith element
            if prev[j] != float('-inf'):
                curr[j] = max(curr[j], prev[j])
 
        # Assigning values of curr to prev list to iterate further
        prev = curr[:]
 
    # Return the answer
    return curr[total_sum]
 
 
# Driver code
if __name__ == "__main__":
    n = 4
    a = [1, 2, 3, 6]
 
    # Function call
    print(max_sum(a, n))

C#

// C# implementation for the above mentioned
// Dynamic Programming approach
 
using System;
using System.Collections.Generic;
 
class GFG
    {
        // Function to find the maximum subset sum
        static int MaxSum(int[] a, int n)
        {
            // sum of all elements
            int sum = 0;
            for (int i = 0; i < n; i++)
                sum += a[i];
 
            int limit = 2 * sum + 1;
 
            // initialising curr and prev vectors table with INT_MIN
            // where, INT_MIN means no solution
            List<int> prev = new List<int>();
            List<int> curr = new List<int>();
 
            for (int i = 0; i <= limit; i++)
            {
                prev.Add(int.MinValue);
                curr.Add(int.MinValue);
            }
             
             // Case when diff is 0
            prev[sum] = 0;
 
            for (int i = 1; i <= n; i++)
            {
                for (int j = 0; j < limit; j++)
                {
                    // Putting ith element in g0
                    if ((j - a[i - 1]) >= 0 && prev[j - a[i - 1]] != int.MinValue)
                        curr[j] = Math.Max(curr[j], prev[j - a[i - 1]] + a[i - 1]);
                     
                    // Putting ith element in g1
                    if ((j + a[i - 1]) < limit && prev[j + a[i - 1]] != int.MinValue)
                        curr[j] = Math.Max(curr[j], prev[j + a[i - 1]]);
                     
                    // Ignoring ith element
                    if (prev[j] != int.MinValue)
                        curr[j] = Math.Max(curr[j], prev[j]);
                }
                 
                // assigning values of curr to prev vector to iterate further
                prev = new List<int>(curr);
            }
             
            // return answer
            return curr[sum];
        }
 
        // Driver code
        static void Main(string[] args)
        {
            int n = 4;
            int[] a = { 1, 2, 3, 6 };
 
            // function call
            Console.WriteLine(MaxSum(a, n));
        }
}
 
// This code is contributed by Vaibhav Nandan

Javascript

function maxSum(a, n) {
    // sum of all elements
    let sum = 0;
    for (let i = 0; i < n; i++)
        sum += a[i];
 
    let limit = 2 * sum + 1;
 
    // initialising curr and prev vectors table with INT_MIN
    // where, INT_MIN means no solution
    let prev = new Array(limit + 1).fill(Number.MIN_SAFE_INTEGER);
    let curr = new Array(limit + 1).fill(Number.MIN_SAFE_INTEGER);
 
    // Case when diff is 0
    prev[sum] = 0;
 
    for (let i = 1; i <= n; i++) {
        for (let j = 0; j < limit; j++) {
 
            // Putting ith element in g0
            if ((j - a[i - 1]) >= 0 && prev[j - a[i - 1]] != Number.MIN_SAFE_INTEGER)
 
                curr[j] = Math.max(curr[j], prev[j - a[i - 1]] + a[i - 1]);
 
            // Putting ith element in g1
            if ((j + a[i - 1]) < limit && prev[j + a[i - 1]] != Number.MIN_SAFE_INTEGER)
 
                curr[j] = Math.max(curr[j], prev[j + a[i - 1]]);
 
            // Ignoring ith element
            if (prev[j] != Number.MIN_SAFE_INTEGER)
 
                curr[j] = Math.max(curr[j], prev[j]);
        }
        // assigning values of curr to prev vector to iterate further
        prev = [...curr];
    }
 
    // return answer
    return curr[sum];
}
 
// Driver code
let n = 4;
let a = [1, 2, 3, 6];
 
// function call
console.log(maxSum(a, n));

Output:

Time Complexity: O(N*Sum)

Auxiliary Space: O(Sum)

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