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Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond

Author Manuel Cáceres

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Manuel Cáceres
  • Department of Computer Science, University of Helsinki, Finland

Funding

This work was partially funded by the Academy of Finland (grants No. 352821, 328877).

Acknowledgements

I am very grateful to Alexandru I. Tomescu for initial discussions on funnel algorithms, to Veli Mäkinen for discussions on applying KMP on DAGs, to Massimo Equi and Nicola Rizzo for the useful discussions, and to the anonymous reviewers for their useful comments.

Cite AsGet BibTex

Manuel Cáceres. Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CPM.2023.7

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 ∈ Σ^m. SMLG can be solved in quadratic O(m|E|) time [Amir et al., JALG 2000], 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 on 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 2020] 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Pattern matching
  • Mathematics of computing → Graph algorithms
Keywords
  • string matching
  • parameterized algorithms
  • FPT inside P
  • string algorithms
  • graph algorithms
  • directed acyclic graphs
  • labeled graphs
  • funnels

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