Question

I have a set S containing natural numbers and a target t which is an number. I want to know
how can we find the number of possible combinations of these numbers which sums up to target t.
A number can be taken any number of times and any number of numbers can be taken for getting the
sum equal to the target t.

Example
target  6
Set s  {3,8,1,2}
Solution   3+3, 3+2+1, 1+1+1+3, 2+2+2, 2+1+1+2, 2+1+1+1+1, 1+1+1+1+1+1
Total No of solutions possible  7

What can be the efficient algorithm for this?

Answer 1

If the target is reasonably small, you can use dynamic programming to obtain an O(|S| * t) solution. Here is a sample code in Python:

def combinations(S, t):
    dp = [1] + [0] * t
    for i in S:
        for j in range(0, t - i + 1):
            dp[j + i] += dp[j]
    return dp[t]