# Partition a set into two subsets such that the difference of subset sums is minimum

Given a set of integers, the task is to divide it into two sets S1 and S2 such that the absolute difference between their sums is minimum.
If there is a set S with n elements, then if we assume Subset1 has m elements, Subset2 must have n-m elements and the value of abs(sum(Subset1) – sum(Subset2)) should be minimum.
Example:

```Input:  arr[] = {1, 6, 11, 5}
Output: 1
Explanation:
Subset1 = {1, 5, 6}, sum of Subset1 = 12
Subset2 = {11}, sum of Subset2 = 11
```

This problem is mainly an extension to the Dynamic Programming| Set 18 (Partition Problem).
Recursive Solution
The recursive approach is to generate all possible sums from all the values of the array and to check which solution is the most optimal one.
To generate sums we either include the i’th item in set 1 or don’t include, i.e., include in set 2.

 `// A Recursive C program to solve minimum sum partition` `// problem.` `#include ` `using` `namespace` `std;`   `// Function to find the minimum sum` `int` `findMinRec(``int` `arr[], ``int` `i, ``int` `sumCalculated, ``int` `sumTotal)` `{` `    ``// If we have reached last element.  Sum of one` `    ``// subset is sumCalculated, sum of other subset is` `    ``// sumTotal-sumCalculated.  Return absolute difference` `    ``// of two sums.` `    ``if` `(i==0)` `        ``return` `abs``((sumTotal-sumCalculated) - sumCalculated);`     `    ``// For every item arr[i], we have two choices` `    ``// (1) We do not include it first set` `    ``// (2) We include it in first set` `    ``// We return minimum of two choices` `    ``return` `min(findMinRec(arr, i-1, sumCalculated+arr[i-1], sumTotal),` `               ``findMinRec(arr, i-1, sumCalculated, sumTotal));` `}`   `// Returns minimum possible difference between sums` `// of two subsets` `int` `findMin(``int` `arr[], ``int` `n)` `{` `    ``// Compute total sum of elements` `    ``int` `sumTotal = 0;` `    ``for` `(``int` `i=0; i

 `// JAVA code to partition a set into two subsets` `// such that the difference of subset sums` `// is minimum` `import` `java.util.*;`   `class` `GFG {` `    `  `    ``// Function to find the minimum sum` `    ``public` `static` `int` `findMinRec(``int` `arr[], ``int` `i,` `                                ``int` `sumCalculated,` `                                 ``int` `sumTotal)` `    ``{` `        ``// If we have reached last element.` `        ``// Sum of one subset is sumCalculated,` `        ``// sum of other subset is sumTotal-` `        ``// sumCalculated.  Return absolute ` `        ``// difference of two sums.` `        ``if` `(i == ``0``)` `            ``return` `Math.abs((sumTotal-sumCalculated) -` `                                   ``sumCalculated);` `     `  `     `  `        ``// For every item arr[i], we have two choices` `        ``// (1) We do not include it first set` `        ``// (2) We include it in first set` `        ``// We return minimum of two choices` `        ``return` `Math.min(findMinRec(arr, i - ``1``, sumCalculated ` `                                   ``+ arr[i-``1``], sumTotal),` `                                 ``findMinRec(arr, i-``1``,` `                                  ``sumCalculated, sumTotal));` `    ``}` `     `  `    ``// Returns minimum possible difference between` `    ``// sums of two subsets` `    ``public` `static` `int` `findMin(``int` `arr[], ``int` `n)` `    ``{` `        ``// Compute total sum of elements` `        ``int` `sumTotal = ``0``;` `        ``for` `(``int` `i = ``0``; i < n; i++)` `            ``sumTotal += arr[i];` `     `  `        ``// Compute result using recursive function` `        ``return` `findMinRec(arr, n, ``0``, sumTotal);` `    ``}` `    `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `main(String[] args) ` `    ``{` `        ``int` `arr[] = {``3``, ``1``, ``4``, ``2``, ``2``, ``1``};` `        ``int` `n = arr.length;` `        ``System.out.print(``"The minimum difference"``+` `                        ``" between two sets is "` `+ ` `                         ``findMin(arr, n));` `    ``}` `}` `  `  `// This code is contributed by Arnav Kr. Mandal.`

 `# Python3 program for the` `# above approach` `# A Recursive C program to ` `# solve minimum sum partition` `# problem.`   `# Function to find the minimum sum` `def` `findMinRec(arr, i, sumCalculated, ` `               ``sumTotal):`   `    ``# If we have reached last element.  ` `    ``# Sum of one subset is sumCalculated, ` `    ``# sum of other subset is sumTotal- ` `    ``# sumCalculated.  Return absolute ` `    ``# difference of two sums.` `    ``if` `(i ``=``=` `0``):` `        ``return` `abs``((sumTotal ``-` `sumCalculated) ``-` `                    ``sumCalculated)`   `    ``# For every item arr[i], we have two choices` `    ``# (1) We do not include it first set` `    ``# (2) We include it in first set` `    ``# We return minimum of two choices` `    ``return` `min``(findMinRec(arr, i ``-` `1``, ` `                          ``sumCalculated``+``arr[i ``-` `1``], ` `                          ``sumTotal),` `               ``findMinRec(arr, i ``-` `1``, ` `                          ``sumCalculated, sumTotal))`   `# Returns minimum possible ` `# difference between sums` `# of two subsets` `def` `findMin(arr,  n):`   `    ``# Compute total sum ` `    ``# of elements` `    ``sumTotal ``=` `0` `    ``for` `i ``in` `range``(n):` `        ``sumTotal ``+``=` `arr[i]`   `    ``# Compute result using ` `    ``# recursive function` `    ``return` `findMinRec(arr, n, ` `                      ``0``, sumTotal)`   `# Driver code` `if` `__name__ ``=``=` `"__main__"``:`   `    ``arr ``=` `[``3``, ``1``, ``4``, ``2``, ``2``, ``1``]` `    ``n ``=` `len``(arr)` `    ``print``(``"The minimum difference "` `+` `          ``"between two sets is "``, ` `           ``findMin(arr, n))`   `# This code is contributed by Chitranayal`

 `// C# code to partition a set into two subsets` `// such that the difference of subset sums` `// is minimum` `using` `System; `   `class` `GFG {` `    `  `    ``// Function to find the minimum sum` `    ``public` `static` `int` `findMinRec(``int` `[]arr, ``int` `i,` `                                ``int` `sumCalculated,` `                                     ``int` `sumTotal)` `    ``{` `        ``// If we have reached last element.` `        ``// Sum of one subset is sumCalculated,` `        ``// sum of other subset is sumTotal-` `        ``// sumCalculated. Return absolute ` `        ``// difference of two sums.` `        ``if` `(i == 0)` `            ``return` `Math.Abs((sumTotal-sumCalculated)` `                                    ``- sumCalculated);` `    `  `    `  `        ``// For every item arr[i], we have two choices` `        ``// (1) We do not include it first set` `        ``// (2) We include it in first set` `        ``// We return minimum of two choices` `        ``return` `Math.Min(findMinRec(arr, i - 1, ` `                  ``sumCalculated + arr[i-1], sumTotal),` `                   ``findMinRec(arr, i-1, sumCalculated, ` `                                           ``sumTotal));` `    ``}` `    `  `    ``// Returns minimum possible difference between` `    ``// sums of two subsets` `    ``public` `static` `int` `findMin(``int` `[]arr, ``int` `n)` `    ``{` `        `  `        ``// Compute total sum of elements` `        ``int` `sumTotal = 0;` `        ``for` `(``int` `i = 0; i < n; i++)` `            ``sumTotal += arr[i];` `    `  `        ``// Compute result using recursive function` `        ``return` `findMinRec(arr, n, 0, sumTotal);` `    ``}` `    `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `Main() ` `    ``{` `        ``int` `[]arr = {3, 1, 4, 2, 2, 1};` `        ``int` `n = arr.Length;` `        ``Console.Write(``"The minimum difference"``+` `                        ``" between two sets is "` `+ ` `                               ``findMin(arr, n));` `    ``}` `}` `    `  `// This code is contributed by nitin mittal.`

Output:

```The minimum difference between two sets is 1

```

Time Complexity:

```All the sums can be generated by either
(1) including that element in set 1.
(2) without including that element in set 1.
So possible combinations are :-
arr      (1 or 2)  -> 2 values
arr    (1 or 2)  -> 2 values
.
.
.
arr[n]     (2 or 2)  -> 2 values
So time complexity will be 2*2*..... *2 (For n times),
that is O(2^n).

```

Dynamic Programming
The problem can be solved using dynamic programming when the sum of the elements is not too big. We can create a 2D array dp[n+1][sum+1] where n is a number of elements in a given set and sum is the sum of all elements. We can construct the solution in a bottom-up manner.

```The task is to divide the set into two parts.
We will consider the following factors for dividing it.
Let
dp[n+1][sum+1] = {1 if some subset from 1st to i'th has a sum
equal to j
0 otherwise}

i ranges from {1..n}
j ranges from {0..(sum of all elements)}

So
dp[n+1][sum+1]  will be 1 if
1) The sum j is achieved including i'th item
2) The sum j is achieved excluding i'th item.

Let sum of all the elements be S.

To find Minimum sum difference, w have to find j such
that Min{sum - j*2  : dp[n][j]  == 1 }
where j varies from 0 to sum/2

The idea is, sum of S1 is j and it should be closest
to sum/2, i.e., 2*j should be closest to sum.
```

Below is the implementation of the above code.

 `// A Recursive C program to solve minimum sum partition` `// problem.` `#include ` `using` `namespace` `std;`   `// Returns the minimum value of the difference of the two sets.` `int` `findMin(``int` `arr[], ``int` `n)` `{` `    ``// Calculate sum of all elements` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``sum += arr[i];`   `    ``// Create an array to store results of subproblems` `    ``bool` `dp[n+1][sum+1];`   `    ``// Initialize first column as true. 0 sum is possible ` `    ``// with all elements.` `    ``for` `(``int` `i = 0; i <= n; i++)` `        ``dp[i] = ``true``;`   `    ``// Initialize top row, except dp, as false. With` `    ``// 0 elements, no other sum except 0 is possible` `    ``for` `(``int` `i = 1; i <= sum; i++)` `        ``dp[i] = ``false``;`   `    ``// Fill the partition table in bottom up manner` `    ``for` `(``int` `i=1; i<=n; i++)` `    ``{` `        ``for` `(``int` `j=1; j<=sum; j++)` `        ``{` `            ``// If i'th element is excluded` `            ``dp[i][j] = dp[i-1][j];`   `            ``// If i'th element is included` `            ``if` `(arr[i-1] <= j)` `                ``dp[i][j] |= dp[i-1][j-arr[i-1]];` `        ``}` `    ``}` ` `  `    ``// Initialize difference of two sums. ` `    ``int` `diff = INT_MAX;` `    `  `    ``// Find the largest j such that dp[n][j]` `    ``// is true where j loops from sum/2 t0 0` `    ``for` `(``int` `j=sum/2; j>=0; j--)` `    ``{` `        ``// Find the ` `        ``if` `(dp[n][j] == ``true``)` `        ``{` `            ``diff = sum-2*j;` `            ``break``;` `        ``}` `    ``}` `    ``return` `diff;` `}`   `// Driver program to test above function` `int` `main()` `{` `    ``int` `arr[] = {3, 1, 4, 2, 2, 1};` `    ``int` `n = ``sizeof``(arr)/``sizeof``(arr);` `    ``cout << ``"The minimum difference between 2 sets is "` `         ``<< findMin(arr, n);` `    ``return` `0;` `}`

 `// A Recursive java program to solve ` `// minimum sum partition problem.` `import` `java.io.*;`   `class` `GFG ` `{` `    ``// Returns the minimum value of ` `    ``//the difference of the two sets.` `    ``static` `int` `findMin(``int` `arr[], ``int` `n)` `    ``{` `        ``// Calculate sum of all elements` `        ``int` `sum = ``0``; ` `        ``for` `(``int` `i = ``0``; i < n; i++)` `            ``sum += arr[i];` `    `  `        ``// Create an array to store ` `        ``// results of subproblems` `        ``boolean` `dp[][] = ``new` `boolean``[n + ``1``][sum + ``1``];` `    `  `        ``// Initialize first column as true. ` `        ``// 0 sum is possible  with all elements.` `        ``for` `(``int` `i = ``0``; i <= n; i++)` `            ``dp[i][``0``] = ``true``;` `    `  `        ``// Initialize top row, except dp, ` `        ``// as false. With 0 elements, no other ` `        ``// sum except 0 is possible` `        ``for` `(``int` `i = ``1``; i <= sum; i++)` `            ``dp[``0``][i] = ``false``;` `    `  `        ``// Fill the partition table ` `        ``// in bottom up manner` `        ``for` `(``int` `i = ``1``; i <= n; i++)` `        ``{` `            ``for` `(``int` `j = ``1``; j <= sum; j++)` `            ``{` `                ``// If i'th element is excluded` `                ``dp[i][j] = dp[i - ``1``][j];` `    `  `                ``// If i'th element is included` `                ``if` `(arr[i - ``1``] <= j)` `                    ``dp[i][j] |= dp[i - ``1``][j - arr[i - ``1``]];` `            ``}` `        ``}` `    `  `        ``// Initialize difference of two sums. ` `        ``int` `diff = Integer.MAX_VALUE;` `        `  `        ``// Find the largest j such that dp[n][j]` `        ``// is true where j loops from sum/2 t0 0` `        ``for` `(``int` `j = sum / ``2``; j >= ``0``; j--)` `        ``{` `            ``// Find the ` `            ``if` `(dp[n][j] == ``true``)` `            ``{` `                ``diff = sum - ``2` `* j;` `                ``break``;` `            ``}` `        ``}` `        ``return` `diff;` `    ``}` `    `  `    ``// Driver program ` `    ``public` `static` `void` `main (String[] args) ` `    ``{` `        ``int` `arr[] = {``3``, ``1``, ``4``, ``2``, ``2``, ``1``};` `        ``int` `n = arr.length;` `        ``System.out.println (``"The minimum difference between 2 sets is "` `                            ``+ findMin(arr, n));` `    `  `    ``}` `}` `// This code is contributed by vt_m`

 `# A Recursive Python3 program to solve` `# minimum sum partition problem.` `import` `sys`   `# Returns the minimum value of the ` `# difference of the two sets.` `def` `findMin(a, n):` `    `  `    ``su ``=` `0` `    `  `    ``# Calculate sum of all elements ` `    ``su ``=` `sum``(a)`   `    ``# Create an 2d list to store ` `    ``# results of subproblems` `    ``dp ``=` `[[``0` `for` `i ``in` `range``(su ``+` `1``)] ` `             ``for` `j ``in` `range``(n ``+` `1``)]`   `    ``# Initialize first column as true.` `    ``# 0 sum is possible ` `    ``# with all elements. ` `    ``for` `i ``in` `range``(n ``+` `1``):` `        ``dp[i][``0``] ``=` `True` `        `  `    ``# Initialize top row, except dp,` `    ``# as false. With 0 elements, no other ` `    ``# sum except 0 is possible` `    ``for` `j ``in` `range``(``1``, su ``+` `1``):` `        ``dp[``0``][j] ``=` `False` `    `  `    ``# Fill the partition table in ` `    ``# bottom up manner` `    ``for` `i ``in` `range``(``1``, n ``+` `1``):` `        ``for` `j ``in` `range``(``1``, su ``+` `1``):` `            `  `            ``# If i'th element is excluded ` `            ``dp[i][j] ``=` `dp[i ``-` `1``][j]` `            `  `            ``# If i'th element is included` `            ``if` `a[i ``-` `1``] <``=` `j:` `                ``dp[i][j] |``=` `dp[i ``-` `1``][j ``-` `a[i ``-` `1``]]` `    `  `    ``# Initialize difference ` `    ``# of two sums.` `    ``diff ``=` `sys.maxsize`   `    ``# Find the largest j such that dp[n][j] ` `    ``# is true where j loops from sum/2 t0 0 ` `    ``for` `j ``in` `range``(su ``/``/` `2``, ``-``1``, ``-``1``):` `        ``if` `dp[n][j] ``=``=` `True``:` `            ``diff ``=` `su ``-` `(``2` `*` `j)` `            ``break` `            `  `    ``return` `diff` `    `  `# Driver code` `a ``=` `[ ``3``, ``1``, ``4``, ``2``, ``2``, ``1` `]` `n ``=` `len``(a)` `    `  `print``(``"The minimum difference between "` `      ``"2 sets is "``, findMin(a, n))`   `# This code is contributed by Tokir Manva`

 `// A Recursive C# program to solve ` `// minimum sum partition problem.` `using` `System;`   `class` `GFG{` `    `  `// Returns the minimum value of ` `//the difference of the two sets.` `static` `int` `findMin(``int` `[]arr, ``int` `n)` `{` `    `  `    ``// Calculate sum of all elements` `    ``int` `sum = 0; ` `    ``for``(``int` `i = 0; i < n; i++)` `        ``sum += arr[i];`   `    ``// Create an array to store ` `    ``// results of subproblems` `    ``bool` `[,]dp = ``new` `bool``[n + 1, sum + 1];`   `    ``// Initialize first column as true. ` `    ``// 0 sum is possible  with all elements.` `    ``for``(``int` `i = 0; i <= n; i++)` `        ``dp[i, 0] = ``true``;`   `    ``// Initialize top row, except dp[0,0], ` `    ``// as false. With 0 elements, no other ` `    ``// sum except 0 is possible` `    ``for``(``int` `i = 1; i <= sum; i++)` `        ``dp[0, i] = ``false``;`   `    ``// Fill the partition table ` `    ``// in bottom up manner` `    ``for``(``int` `i = 1; i <= n; i++)` `    ``{` `        ``for``(``int` `j = 1; j <= sum; j++)` `        ``{` `            `  `            ``// If i'th element is excluded` `            ``dp[i, j] = dp[i - 1, j];`   `            ``// If i'th element is included` `            ``if` `(arr[i - 1] <= j)` `                ``dp[i, j] |= dp[i - 1, j - arr[i - 1]];` `        ``}` `    ``}`   `    ``// Initialize difference of two sums. ` `    ``int` `diff = ``int``.MaxValue;` `    `  `    ``// Find the largest j such that dp[n,j]` `    ``// is true where j loops from sum/2 t0 0` `    ``for``(``int` `j = sum / 2; j >= 0; j--)` `    ``{` `        `  `        ``// Find the ` `        ``if` `(dp[n, j] == ``true``)` `        ``{` `            ``diff = sum - 2 * j;` `            ``break``;` `        ``}` `    ``}` `    ``return` `diff;` `}`   `// Driver code` `public` `static` `void` `Main(String[] args) ` `{` `    ``int` `[]arr = { 3, 1, 4, 2, 2, 1 };` `    ``int` `n = arr.Length;` `    `  `    ``Console.WriteLine(``"The minimum difference "` `+` `                      ``"between 2 sets is "` `+ ` `                      ``findMin(arr, n));` `}` `}`   `// This code is contributed by Rajput-Ji`

Output:

```The minimum difference between 2 sets is 1

```

Time Complexity = O(n*sum) where n is the number of elements and sum is the sum of all elements.

Note that the above solution is in Pseudo Polynomial Time (time complexity is dependent on the numeric value of input).