The Labeled Direct Product Optimally Solves String Problems on Graphs

TL;DR

This paper introduces the labeled direct product of graphs as a universal tool to develop optimal, efficient algorithms for string problems on labeled graphs, including some previously open problems.

Contribution

The paper presents the labeled direct product as a general framework for solving string problems on labeled graphs, simplifying algorithms and extending solutions to new cases.

Findings

Algorithms for string matching and longest common substring are linear in the size of the labeled product graph.

The approach solves the open problem of LCSP on graphs with cycles.

The algorithms are proven optimal under the Orthogonal Vectors Hypothesis.

Abstract

Suffix trees are an important data structure at the core of optimal solutions to many fundamental string problems, such as exact pattern matching, longest common substring, matching statistics, and longest repeated substring. Recent lines of research focused on extending some of these problems to vertex-labeled graphs, although using ad-hoc approaches which in some cases do not generalize to all input graphs. In the absence of a ubiquitous tool like the suffix tree for labeled graphs, we introduce the labeled direct product of two graphs as a general tool for obtaining optimal algorithms: we obtain conceptually simpler algorithms for the quadratic problems of string matching (SMLG) and longest common substring (LCSP) in labeled graphs. Our algorithms are also more efficient, since they run in time linear in the size of the labeled product graph, which may be smaller than quadratic for some inputs, and their run-time is predictable, because the size of the labeled direct product graph can be precomputed efficiently. We also solve LCSP on graphs containing cycles, which was left as an open problem by Shimohira et al. in 2011. To show the power of the labeled product graph, we also apply it to solve the matching statistics (MSP) and the longest repeated string (LRSP) problems in labeled graphs. Moreover, we show that our (worst-case quadratic) algorithms are also optimal, conditioned on the Orthogonal Vectors Hypothesis. Finally, we complete the complexity picture around LRSP by studying it on undirected graphs.