The Complexity of Unary Subset Sum

Abstract

Given a stream of n numbers and a number B, the subset sum problem deals with checking whether there exists a subset of the stream that adds to exactly B. The unary subset sum problem, USS, is the same problem when the input is encoded in unary. We prove that any p-pass randomized algorithm computing USS with error at most 1/3 must use space \(\Omega(\frac{B}{p})\). For p ≤ B, we give a randomized p-pass algorithm that computes USS with error at most 1/3 using space \(\tilde{O}(\frac{nB}{p})\). We give a deterministic one-pass algorithm which given an input stream and two parameters B, ε, decides whether there exist a subset of the input stream that adds to a value in the range \(\left[(1-\epsilon)B,(1+\epsilon)B\right]\) using space \(O\left(\frac{\log B}{\epsilon}\right)\). We observe that USS is monotone (under a suitable encoding) and give a monotone NC² circuit for USS. We also show that any circuit using ε-approximator gates for USS under this encoding needs Ω(n/logn) gates to compute the Disjointness function.