Fast Pattern Matching with Epsilon Transitions

TL;DR

This paper extends efficient pattern matching algorithms on Wheeler automata to include epsilon transitions, significantly enhancing their expressive power while maintaining linear time and space complexity.

Contribution

It introduces a method to incorporate epsilon transitions into Wheeler automata-based pattern matching without increasing complexity, using only two additional bitvectors.

Findings

Pattern matching with epsilon transitions is achievable in linear time.

Two additional bitvectors suffice to handle epsilon transitions efficiently.

The approach maintains space efficiency comparable to previous models.

Abstract

In the String Matching in Labeled Graphs (SMLG) problem, we need to determine whether a pattern string appears on a given labeled graph or a given automaton. Under the Orthogonal Vectors hypothesis, the SMLG problem cannot be solved in subquadratic time [ICALP 2019]. In typical bioinformatics applications, pattern matching algorithms should be both fast and space-efficient, so we need to determine useful classes of graphs on which the SLMG problem can be solved efficiently. In this paper, we improve on a recent result [STACS 2024] that shows how to solve the SMLG problem in linear time on the compressed representation of Wheeler generalized automata, a class of string-labeled automata that extend de Bruijn graphs. More precisely, we show how to remove the assumption that the automata contain no $ \epsilon $-transitions (namely, edges labeled with the empty string), while retaining the same time and space bounds. This is a significant improvement because $ \epsilon $-transitions add considerable expressive power (making it possible to jump to multiple states for free) and capture the complexity of regular expressions (through Thompson's construction for converting a regular expression into an equivalent automaton). We prove that, to enable $ \epsilon $-transitions, we only need to store two additional bitvectors that can be constructed in linear time.