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Dynamic Data Structures for Parameterized String Problems

Authors Jędrzej Olkowski, Michał Pilipczuk , Mateusz Rychlicki , Karol Węgrzycki , Anna Zych-Pawlewicz

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Author Details

Jędrzej Olkowski
  • Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Mateusz Rychlicki
  • School of Computing, University of Leeds, UK
Karol Węgrzycki
  • Saarland University, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Anna Zych-Pawlewicz
  • Institute of Informatics, University of Warsaw, Poland

Funding

This work is the result of research conducted within research project number 2017/26/D/ST6/00264 financed by National Science Centre, Poland (Jędrzej Olkowski and Anna Zych-Pawlewicz). This work is a part of projects BOBR (Michał Pilipczuk) and TIPEA (Karol Węgrzycki) that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements no. 948057 and 850979, respectively). Mateusz Rychlicki acknowledges support from the Engineering and Physical Sciences Research Council (EPSRC, project EP/V00252X/1).

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Jędrzej Olkowski, Michał Pilipczuk, Mateusz Rychlicki, Karol Węgrzycki, and Anna Zych-Pawlewicz. Dynamic Data Structures for Parameterized String Problems. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 50:1-50:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.50

Abstract

We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the CLOSEST STRING problem, for which we design randomized dynamic data structures with amortized update times d^𝒪(d) and |Σ|^𝒪(d), respectively, where Σ is the alphabet and d is the assumed bound on the maximum distance. These are obtained by combining known static approaches to CLOSEST STRING with color-coding. Next, we note that from a result of Frandsen et al. [J. ACM'97] one can easily infer a meta-theorem that provides dynamic data structures for parameterized string problems with worst-case update time of the form 𝒪_k(log log n), where k is the parameter in question and n is the length of the string. We showcase the utility of this meta-theorem by giving such data structures for problems DISJOINT FACTORS and EDIT DISTANCE. We also give explicit data structures for these problems, with worst-case update times 𝒪(k 2^k log log n) and 𝒪(k²log log n), respectively. Finally, we discuss how a lower bound methodology introduced by Amarilli et al. [ICALP'21] can be used to show that obtaining update time 𝒪(f(k)) for DISJOINT FACTORS and EDIT DISTANCE is unlikely already for a constant value of the parameter k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Predecessor queries
Keywords
  • Parameterized algorithms
  • Dynamic data structures
  • String problems
  • Closest String
  • Edit Distance
  • Disjoint Factors
  • Predecessor problem

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