# Count ways to split array into two subsets having difference between their sum equal to K

Given an array A[] of size N and an integer diff, the task is to count the number of ways to split the array into two subsets (non-empty subset is possible) such that the difference between their sums is equal to diff.

Examples:

Input: A[] = {1, 1, 2, 3}, diff = 1
Output: 3
Explanation: All possible combinations are as follows:

• {1, 1, 2} and {3}
• {1, 3} and {1, 2}
• {1, 2} and {1, 3}

All partitions have difference between their sums equal to 1. Therefore, the count of ways is 3.

Input: A[] = {1, 6, 11, 5}, diff=1
Output: 2

Naive Approach: The simplest approach to solve the problem is based on the following observations:

Let the sum of elements in the partition subsets S1 and S2 be sum1 and sum2 respectively.
Let sum of the array A[] be X
Given, sum1 – sum2 = diff – (1)
Also, sum1 + sum2 = X – (2)

From equations (1) and (2),
sum1 = (X + diff)/2

Therefore, the task is reduced to finding the number of subsets with a given sum
Therefore, the simplest approach is to solve this problem is by generating all the possible subsets and checking whether the subset has the required sum.

Time Complexity: O(2N)
Auxiliary Space: O(N)

Efficient Approach: To optimize the above approach, the idea is to use Dynamic Programming. Initialize a dp[][] table of size N*X, where dp[i][C] stores the number of subsets of the sub-array A[iâ€¦N-1] such that their sum is equal to C. Thus, the recurrence is very trivial as there are only two choices i.e. either consider the ith element in the subset or donâ€™t. So the recurrence relation will be:

dp[i][C] = dp[i – 1][C – A[i]] + dp[i-1][C]

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function to count the number of ways to divide``// the array into two subsets and such that the``// difference between their sums is equal to diff``int` `countSubset(``int` `arr[], ``int` `n, ``int` `diff)``{``    ``// Store the sum of the set S1``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``        ``sum += arr[i];``    ``sum += diff;``    ``sum = sum / 2;` `    ``// Initializing the matrix``    ``int` `t[n + 1][sum + 1];` `    ``// Number of ways to get sum``    ``// using 0 elements is 0``    ``for` `(``int` `j = 0; j <= sum; j++)``        ``t[0][j] = 0;` `    ``// Number of ways to get sum 0``    ``// using i elements is 1``    ``for` `(``int` `i = 0; i <= n; i++)``        ``t[i][0] = 1;` `    ``// Traverse the 2D array``    ``for` `(``int` `i = 1; i <= n; i++) {``        ``for` `(``int` `j = 1; j <= sum; j++) {` `            ``// If the value is greater``            ``// than the sum store the``            ``// value of previous state``            ``if` `(arr[i - 1] > j)``                ``t[i][j] = t[i - 1][j];` `            ``else` `{``                ``t[i][j] = t[i - 1][j]``                          ``+ t[i - 1][j - arr[i - 1]];``            ``}``        ``}``    ``}` `    ``// Return the result``    ``return` `t[n][sum];``}` `// Driver Code``int` `main()``{``    ``// Given Input``    ``int` `diff = 1, n = 4;``    ``int` `arr[] = { 1, 1, 2, 3 };` `    ``// Function Call``    ``cout << countSubset(arr, n, diff);``}`

## Java

 `// Java program for the above approach``import` `java.io.*;``public` `class` `GFG``{` `    ``// Function to count the number of ways to divide``    ``// the array into two subsets and such that the``    ``// difference between their sums is equal to diff``    ``static` `int` `countSubset(``int` `[]arr, ``int` `n, ``int` `diff)``    ``{``      ` `        ``// Store the sum of the set S1``        ``int` `sum = ``0``;``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``sum += arr[i];``        ``sum += diff;``        ``sum = sum / ``2``;``    ` `        ``// Initializing the matrix``        ``int` `t[][] = ``new` `int``[n + ``1``][sum + ``1``];``    ` `        ``// Number of ways to get sum``        ``// using 0 elements is 0``        ``for` `(``int` `j = ``0``; j <= sum; j++)``            ``t[``0``][j] = ``0``;``    ` `        ``// Number of ways to get sum 0``        ``// using i elements is 1``        ``for` `(``int` `i = ``0``; i <= n; i++)``            ``t[i][``0``] = ``1``;``    ` `        ``// Traverse the 2D array``        ``for` `(``int` `i = ``1``; i <= n; i++) {``            ``for` `(``int` `j = ``1``; j <= sum; j++) {``    ` `                ``// If the value is greater``                ``// than the sum store the``                ``// value of previous state``                ``if` `(arr[i - ``1``] > j)``                    ``t[i][j] = t[i - ``1``][j];``    ` `                ``else` `{``                    ``t[i][j] = t[i - ``1``][j]``                              ``+ t[i - ``1``][j - arr[i - ``1``]];``                ``}``            ``}``        ``}``    ` `        ``// Return the result``        ``return` `t[n][sum];``    ``}``    ` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ` `        ``// Given Input``        ``int` `diff = ``1``, n = ``4``;``        ``int` `arr[] = { ``1``, ``1``, ``2``, ``3` `};``    ` `        ``// Function Call``        ``System.out.print(countSubset(arr, n, diff));``    ``}``}` `// This code is contributed by AnkThon`

## Python3

 `# Python3 program for the above approach` `# Function to count the number of ways to divide``# the array into two subsets and such that the``# difference between their sums is equal to diff``def` `countSubset(arr, n, diff):``    ` `    ``# Store the sum of the set S1``    ``sum` `=` `0``    ``for` `i ``in` `range``(n):``        ``sum` `+``=` `arr[i]``        ` `    ``sum` `+``=` `diff``    ``sum` `=` `sum` `/``/` `2` `    ``# Initializing the matrix``    ``t ``=` `[[``0` `for` `i ``in` `range``(``sum` `+` `1``)] ``            ``for` `i ``in` `range``(n ``+` `1``)]` `    ``# Number of ways to get sum``    ``# using 0 elements is 0``    ``for` `j ``in` `range``(``sum` `+` `1``):``        ``t[``0``][j] ``=` `0` `    ``# Number of ways to get sum 0``    ``# using i elements is 1``    ``for` `i ``in` `range``(n ``+` `1``):``        ``t[i][``0``] ``=` `1` `    ``# Traverse the 2D array``    ``for` `i ``in` `range``(``1``, n ``+` `1``):``        ``for` `j ``in` `range``(``1``, ``sum` `+` `1``):``            ` `            ``# If the value is greater``            ``# than the sum store the``            ``# value of previous state``            ``if` `(arr[i ``-` `1``] > j):``                ``t[i][j] ``=` `t[i ``-` `1``][j]``            ``else``:``                ``t[i][j] ``=` `t[i ``-` `1``][j] ``+` `t[i ``-` `1``][j ``-` `arr[i ``-` `1``]]` `    ``# Return the result``    ``return` `t[n][``sum``]` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``# Given Input``    ``diff, n ``=` `1``, ``4``    ``arr ``=` `[ ``1``, ``1``, ``2``, ``3` `]` `    ``# Function Call``    ``print` `(countSubset(arr, n, diff))` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program for the above approach` `using` `System;` `public` `class` `GFG``{` `    ``// Function to count the number of ways to divide``    ``// the array into two subsets and such that the``    ``// difference between their sums is equal to diff``    ``static` `int` `countSubset(``int` `[]arr, ``int` `n, ``int` `diff)``    ``{``      ` `        ``// Store the sum of the set S1``        ``int` `sum = 0;``        ``for` `(``int` `i = 0; i < n; i++)``            ``sum += arr[i];``        ``sum += diff;``        ``sum = sum / 2;``    ` `        ``// Initializing the matrix``        ``int` `[,]t = ``new` `int``[n + 1, sum + 1];``    ` `        ``// Number of ways to get sum``        ``// using 0 elements is 0``        ``for` `(``int` `j = 0; j <= sum; j++)``            ``t[0,j] = 0;``    ` `        ``// Number of ways to get sum 0``        ``// using i elements is 1``        ``for` `(``int` `i = 0; i <= n; i++)``            ``t[i,0] = 1;``    ` `        ``// Traverse the 2D array``        ``for` `(``int` `i = 1; i <= n; i++) {``            ``for` `(``int` `j = 1; j <= sum; j++) {``    ` `                ``// If the value is greater``                ``// than the sum store the``                ``// value of previous state``                ``if` `(arr[i - 1] > j)``                    ``t[i,j] = t[i - 1,j];``    ` `                ``else` `{``                    ``t[i,j] = t[i - 1,j]``                              ``+ t[i - 1,j - arr[i - 1]];``                ``}``            ``}``        ``}``    ` `        ``// Return the result``        ``return` `t[n,sum];``    ``}``    ` `    ``// Driver Code``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ` `        ``// Given Input``        ``int` `diff = 1, n = 4;``        ``int` `[]arr = { 1, 1, 2, 3 };``    ` `        ``// Function Call``        ``Console.Write(countSubset(arr, n, diff));``    ``}``}` `// This code is contributed by AnkThon`

## Javascript

 ``

Output:
`3`

Time Complexity: O(S*N), where S = sum of array elements + K/2
Auxiliary Space: O(S*N)

Efficient approach: space optimization

To optimize space complexity we only need to keep track of the values of the previous row in the 2D array to compute the values of the current row. Hence, we can replace the 2D array t[n + 1][sum + 1] with a 1D array dp[sum + 1].

Implementation Steps :

• Create vector Dp of size sum + 1 and initialize it with 0.
• Now initialize DP with Base Case dp[0] =1.
• To calculate answer iterate over subproblems with the help of nested loops and get the current value from previous computation.
• At last return the answer stored in dp[sum].

Implementation:

## C++

 `// C++ program for above approach` `#include ``using` `namespace` `std;`  `// Function to count the number of ways to divide``// the array into two subsets and such that the``// difference between their sums is equal to diff``int` `countSubset(``int` `arr[], ``int` `n, ``int` `diff)``{ ``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``        ``sum += arr[i];``    ``sum += diff;``    ``sum = sum / 2;``     ` `      ``// Initializing the vector Dp``    ``int` `dp[sum + 1] = {0};``  ` `      ``// Base Case``    ``dp[0] = 1;``     ` `      ``// iterate over subproblems to get the current computation``    ``for` `(``int` `i = 0; i < n; i++) {``        ``for` `(``int` `j = sum; j >= arr[i]; j--) {``              ``// update DP from previous values``            ``dp[j] += dp[j - arr[i]];``        ``}``    ``}``     ` `      ``// return answer``    ``return` `dp[sum];``}` `// Driver Code``int` `main()``{``    ``// Given Input``    ``int` `diff = 1, n = 4;``    ``int` `arr[] = { 1, 1, 2, 3 };` `    ``// Function Call``    ``cout << countSubset(arr, n, diff);``}` `// this code is contributed by bhardwajji`

## Java

 `// Java program for above approach``import` `java.util.*;` `public` `class` `Main``{``  ` `    ``// Function to count the number of ways to divide``    ``// the array into two subsets and such that the``    ``// difference between their sums is equal to diff``    ``public` `static` `int` `countSubset(``int` `arr[], ``int` `n, ``int` `diff)``    ``{``        ``int` `sum = ``0``;``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``sum += arr[i];``        ``sum += diff;``        ``sum = sum / ``2``;` `        ``// Initializing the vector Dp``        ``int` `dp[] = ``new` `int``[sum + ``1``];` `        ``// Base Case``        ``dp[``0``] = ``1``;` `        ``// iterate over subproblems to get the current computation``        ``for` `(``int` `i = ``0``; i < n; i++) {``            ``for` `(``int` `j = sum; j >= arr[i]; j--) {``                ``// update DP from previous values``                ``dp[j] += dp[j - arr[i]];``            ``}``        ``}` `        ``// return answer``        ``return` `dp[sum];``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``// Given Input``        ``int` `diff = ``1``, n = ``4``;``        ``int` `arr[] = { ``1``, ``1``, ``2``, ``3` `};` `        ``// Function Call``        ``System.out.println(countSubset(arr, n, diff));``    ``}``}`

## Python3

 `# Function to count the number of ways to divide``# the array into two subsets and such that the``# difference between their sums is equal to diff`  `def` `countSubset(arr, n, diff):``    ``# Calculating the sum of all elements``    ``sum` `=` `0``    ``for` `i ``in` `range``(n):``        ``sum` `+``=` `arr[i]``    ``sum` `+``=` `diff` `    ``# If sum is odd, then no such subset can exist``    ``if` `sum` `%` `2` `!``=` `0``:``        ``return` `0` `    ``# Initializing the vector Dp``    ``dp ``=` `[``0``] ``*` `(``sum` `/``/` `2` `+` `1``)` `    ``# Base Case``    ``dp[``0``] ``=` `1` `    ``# iterate over subproblems to get the current computation``    ``for` `i ``in` `range``(n):``        ``for` `j ``in` `range``(``sum` `/``/` `2``, arr[i] ``-` `1``, ``-``1``):``            ``# update DP from previous values``            ``dp[j] ``+``=` `dp[j ``-` `arr[i]]` `    ``# return answer``    ``return` `dp[``sum` `/``/` `2``]`  `# Given Input``diff ``=` `1``n ``=` `4``arr ``=` `[``1``, ``1``, ``2``, ``3``]` `# Function Call``print``(countSubset(arr, n, diff))`

## C#

 `using` `System;` `class` `MainClass``{``  ` `  ``// Function to count the number of ways to divide``  ``// the array into two subsets and such that the``  ``// difference between their sums is equal to diff``  ``static` `int` `countSubset(``int``[] arr, ``int` `n, ``int` `diff)``  ``{``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++) {``      ``sum += arr[i];``    ``}``    ``sum += diff;``    ``sum = sum / 2;` `    ``// Initializing the vector Dp``    ``int``[] dp = ``new` `int``[sum + 1];` `    ``// Base Case``    ``dp[0] = 1;` `    ``// Iterate over subproblems to get the current``    ``// computation``    ``for` `(``int` `i = 0; i < n; i++) {``      ``for` `(``int` `j = sum; j >= arr[i]; j--) {``        ``// Update DP from previous values``        ``dp[j] += dp[j - arr[i]];``      ``}``    ``}` `    ``// Return answer``    ``return` `dp[sum];``  ``}` `  ``// Driver Code``  ``public` `static` `void` `Main()``  ``{``    ``// Given Input``    ``int` `diff = 1, n = 4;``    ``int``[] arr = { 1, 1, 2, 3 };` `    ``// Function Call``    ``Console.WriteLine(countSubset(arr, n, diff));``  ``}``}`

## Javascript

 `// Function to count the number of ways to divide``// the array into two subsets and such that the``// difference between their sums is equal to diff``function` `countSubset(arr, n, diff) {``  ``let sum = 0;``  ``for` `(let i = 0; i < n; i++) {``    ``sum += arr[i];``  ``}``  ``sum += diff;``  ``sum = Math.floor(sum / 2);` `  ``// Initializing the vector Dp``  ``let dp = ``new` `Array(sum + 1).fill(0);` `  ``// Base Case``  ``dp[0] = 1;` `  ``// Iterate over subproblems to get the current``  ``// computation``  ``for` `(let i = 0; i < n; i++) {``    ``for` `(let j = sum; j >= arr[i]; j--) {``      ``// Update DP from previous values``      ``dp[j] += dp[j - arr[i]];``    ``}``  ``}` `  ``// Return answer``  ``return` `dp[sum];``}` `// Driver Code``let diff = 1,``  ``n = 4;``let arr = [1, 1, 2, 3];` `// Function Call``console.log(countSubset(arr, n, diff));`

Output
`3`

Time Complexity: O(S*N), where S = sum of array elements + K/2
Auxiliary Space: O(S)

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