Dictionary matching with one gap

The dictionary matching with gaps problem is to preprocess a dictionary D of d gapped patterns P ₁,...,P _d over alphabet ∑, where each gapped pattern P _i is a sequence of subpatterns separated by bounded sequences of don't cares. Then, given a query text T of length n over alphabet ∑, the goal is to output all locations in T in which a pattern P_i ∈ D, 1 ≤ I ≤ d, ends. There is a renewed current interest in the gapped matching problem stemming from cyber security. In this paper we solve the problem where all patterns in the dictionary have one gap with at least α and at most β don't cares, where α and β are given parameters. Specifically, we show that the dictionary matching with a single gap problem can be solved in either O(d log d+D) time and O(dlog ^ε d+D) space, and query time O(n(β-α)loglogd log ² min { d, log D }+occ), where occ is the number of patterns found, or preprocessing time: O(d² ovr+ D ), where ovr is the maximal number of subpatterns including each other as a prefix or as a suffix, space: O(d ²+ D ), and query time O(n(β-α)+occ), where occ is the number of patterns found. As far as we know, this is the best solution for this setting of the problem, where many overlaps may exist in the dictionary.