Count of subsets with sum equal to target using Recursion

Given an array arr[] of length n and an integer target, the task is to find the number of subsets with a sum equal to target.

Examples:

Input: arr[] = [1, 2, 3, 3], target = 6 
Output: 3 
Explanation: All the possible subsets are [1, 2, 3], [1, 2, 3] and [3, 3]

Input: arr[] = [1, 1, 1, 1], target = 1 
Output: 4 
Explanation: All the possible subsets are [1], [1], [1] and [1]

Approach:

We can identify a recursive pattern in this problem. There are two state variables:

1. The current index i in the array arr[].
2. The cumulative sum currentSum of the subsets being considered.

We consider two cases for recursion:

Exclude the current element: The current element is not included in the subset, and the sum remains unchanged. This corresponds to:

• countSubsets(i + 1, currentSum, target).

Include the current element: The current element is included in the subset, and the sum is updated as currentSum + arr[i]. This corresponds to:

• countSubsets(i + 1, currentSum + arr[i], target)

Recurrence relation:

• countSubsets(i, currentSum, target) = countSubsets(i+1, currentSum, target) + countSubsets(i+1, currentSum + arr[i], target)​

Base Case: If i == n (all elements have been processed), return 1 if the currentSumequal to target else 0.

3

Time Complexity: O(2^n), where n is the size of array.
Auxiliary Space: O(n)

Please refer to efficient method to solve the problem using Dynamic Programming.