Fine-Grained Complexity of Regular Expression Pattern Matching and Membership

TL;DR

This paper investigates the fine-grained complexity of regular expression pattern matching for homogeneous patterns of bounded depth, establishing a dichotomy between pattern types that allow significant improvements and those limited to minor log-factor enhancements.

Contribution

It provides a detailed classification and complexity analysis of homogeneous regular expression patterns, identifying which types admit super-poly-logarithmic improvements and which are constrained under SETH and Formula-SAT hypotheses.

Findings

Few pattern types allow super-poly-logarithmic improvements.

Most pattern types can only be improved by a constant number of log-factors.

The results depend on the hardness assumptions like SETH and Formula-SAT.

Abstract

The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of bounded depth in this work. For them a classification splitting the types in easy (strongly sub-quadratic) and hard (essentially quadratic time under SETH) is known. We take a very fine-grained look at the hard pattern types from this classification and show a dichotomy: few types allow super-poly-logarithmic improvements while the algorithms for the other pattern types can only be improved by a constant number of log-factors, assuming the Formula-SAT Hypothesis.