Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond

TL;DR

This paper introduces parameterized algorithms for string matching in DAGs, focusing on funnel structures and their generalizations, leveraging a generalized KMP algorithm to improve efficiency based on graph topology.

Contribution

It presents the first parameterized algorithms for string matching in DAGs, utilizing graph structure parameters and generalizing funnel classes for improved performance.

Findings

Algorithms are optimized for topological graph parameters.

Generalization of funnel structures to k-funnels and ST_k classes.

Results improve understanding of string matching complexity in DAGs.

Abstract

The problem of String Matching to Labeled Graphs (SMLG) asks to find all the paths in a labeled graph $G = (V, E)$ whose spellings match that of an input string $S \in \Sigma^m$. SMLG can be solved in quadratic $O(m|E|)$ time [Amir et al., JALG], which was proven to be optimal by a recent lower bound conditioned on SETH [Equi et al., ICALP 2019]. The lower bound states that no strongly subquadratic time algorithm exists, even if restricted to directed acyclic graphs (DAGs). In this work we present the first parameterized algorithms for SMLG in DAGs. Our parameters capture the topological structure of $G$. All our results are derived from a generalization of the Knuth-Morris-Pratt algorithm [Park and Kim, CPM 1995] optimized to work in time proportional to the number of prefix-incomparable matches. To obtain the parameterization in the topological structure of $G$, we first study a special class of DAGs called funnels [Millani et al., JCO] and generalize them to $k$-funnels and the class $ST_k$. We present several novel characterizations and algorithmic contributions on both funnels and their generalizations.