Finding submasses in weighted strings with Fast Fourier Transform

We study the Submass Finding Problem: Given a string s over a weighted alphabet, i.e., an alphabet Σ with a weight function μ : Σ → N, we refer to a mass M ∈ N as a submass of s if s has a substring whose weights sum up to M . Now, for a set of input masses {M1, . . . ,Mk}, we want to find those Mi which are submasses of s, and return one or all occurrences of substrings with mass Mi. We present efficient algorithms for both the decision and the search problem. Furthermore, our approach allows us to compute efficiently the number of different submasses of s. The main idea of our algorithms is to define appropriate polynomials such that we can determine the solution for the Submass Finding Problem from the coefficients of the product of these polynomials. We obtain very efficient running times by using Fast Fourier Transform to compute this product. Most of our algorithms run in time O(μs logμs), where μs is the total mass of string s. Employing methods for compressing sparse polynomials, this runtime can be viewed as O(σ(s) log σ(s)), where σ(s) denotes the number of different submasses of s. This allows us to obtain algorithms that have running time as a function of σ(s). Our algorithms give the best known running times, if σ(s) is small. However, in general σ(s) could be as large as O(n2), in which case our algorithms are not necessarily optimal.