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Pattern Matching with Mismatches and Wildcards

Authors Gabriel Bathie , Panagiotis Charalampopoulos , Tatiana Starikovskaya

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Author Details

Gabriel Bathie
  • DIENS, École normale supérieure de Paris, PSL Research University, France
  • LaBRI, Université de Bordeaux, Talence, France
Panagiotis Charalampopoulos
  • Birkbeck, University of London, UK
Tatiana Starikovskaya
  • DIENS, École normale supérieure de Paris, PSL Research University, France

Funding

A research visit during which part of the presented ideas were conceived was funded by a Royal Society International Exchanges Award.
  • Bathie, Gabriel: Partially supported by the grant ANR-20-CE48-0001 from the French National Research Agency (ANR).
  • Starikovskaya, Tatiana: Partially supported by the grant ANR-20-CE48-0001 from the French National Research Agency (ANR).

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Gabriel Bathie, Panagiotis Charalampopoulos, and Tatiana Starikovskaya. Pattern Matching with Mismatches and Wildcards. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.20

Abstract

In this work, we address the problem of approximate pattern matching with wildcards. Given a pattern P of length m containing D wildcards, a text T of length n, and an integer k, our objective is to identify all fragments of T within Hamming distance k from P.
Our primary contribution is an algorithm with runtime 𝒪(n + (D+k)(G+k)⋅ n/m) for this problem. Here, G ≤ D represents the number of maximal wildcard fragments in P. We derive this algorithm by elaborating in a non-trivial way on the ideas presented by [Charalampopoulos, Kociumaka, and Wellnitz, FOCS'20] for pattern matching with mismatches (without wildcards). Our algorithm improves over the state of the art when D, G, and k are small relative to n. For instance, if m = n/2, k = G = n^{2/5}, and D = n^{3/5}, our algorithm operates in 𝒪(n) time, surpassing the Ω(n^{6/5}) time requirement of all previously known algorithms.
In the case of exact pattern matching with wildcards (k = 0), we present a much simpler algorithm with runtime 𝒪(n + DG ⋅ n/m) that clearly illustrates our main technical innovation: the utilisation of positions of P that do not belong to any fragment of P with a density of wildcards much larger than D/m as anchors for the sought (approximate) occurrences. Notably, our algorithm outperforms the best-known 𝒪(n log m)-time FFT-based algorithms of [Cole and Hariharan, STOC'02] and [Clifford and Clifford, IPL'04] if DG = o(m log m).
We complement our algorithmic results with a structural characterization of the k-mismatch occurrences of P. We demonstrate that in a text of length 𝒪(m), these occurrences can be partitioned into 𝒪((D+k)(G+k)) arithmetic progressions. Additionally, we construct an infinite family of examples with Ω((D+k)k) arithmetic progressions of occurrences, leveraging a combinatorial result on progression-free sets [Elkin, SODA'10].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • pattern matching
  • wildcards
  • mismatches
  • Hamming distance

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