Does a combination of K integers exist, so that their sum is equal to a given number?

Question

I've been breaking a sweat over this question I've been asked to answer (it's technically homework). I've considered a hashtable but I'm kind of stuck on the exact specifics of how I'd make this work

Here's the question:

Given k sets of integers A₁,A₂,..,A_k of total size O(n), you should determine whether exist a₁ ϵ A₁, a₂ ϵ A₂,..,a_k ϵ A_k, such that a₁+a₂+..+a_k−1 =a_k. Your algorithm should run in T_k(n) time, where T_k(n) = O(n^k/2 × log n) for even k, and O(n^(k+1)/2) for odd values of k.

Can anyone give me a general direction so that I can come closer to solving this?

Answer 1

Divide the k sets into two groups. For even k, both groups have k/2 sets each. For odd k, one group has (k+1)/2 and the other has (k-1)/2 sets. Compute all possible sums (taking one element from each set) within each group. For even k, you will get two arrays, each with n^k/2 elements. For odd k, one array has n^(k+1)/2 and the other array has n^(k-1)/2 elements. The problem is reduced to the standard one "Given two arrays, check if a specified sum can be reached by taking one element from each array".