Fast Matching of Regular Patterns with Synchronizing Counting

Fast matching of regular expressions with bounded repetition, aka counting, such as (ab){50,100}, i.e., matching linear in the length of the text and independent of the repetition bounds, has been an open problem for at least two decades. We show that, for a wide class of regular expressions with counting, which we call synchronizing, fast matching is possible. We empirically show that the class covers nearly all counting used in usual applications of regex matching. This complexity result is based on an improvement and analysis of a recent matching algorithm that compiles regexes to deterministic counting-set automata (automata with registers that hold sets of numbers).