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Subsequences with Generalised Gap Constraints: Upper and Lower Complexity Bounds

Authors Florin Manea , Jonas Richardsen, Markus L. Schmid

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Author Details

Florin Manea
  • Computer Science Department and CIDAS, Universität Göttingen, Germany
Jonas Richardsen
  • Computer Science Department and CIDAS, Universität Göttingen, Germany
Markus L. Schmid
  • Humboldt-Universität zu Berlin, Berlin, Germany

Funding

  • Manea, Florin: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) in the Heisenberg programme, project number 466789228.
  • Schmid, Markus L.: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) – project number 522576760 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 522576760).

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Florin Manea, Jonas Richardsen, and Markus L. Schmid. Subsequences with Generalised Gap Constraints: Upper and Lower Complexity Bounds. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CPM.2024.22

Abstract

For two strings u, v over some alphabet A, we investigate the problem of embedding u into w as a subsequence under the presence of generalised gap constraints. A generalised gap constraint is a triple (i, j, C_{i, j}), where 1 ≤ i < j ≤ |u| and C_{i, j} ⊆ A^*. Embedding u as a subsequence into v such that (i, j, C_{i, j}) is satisfied means that if u[i] and u[j] are mapped to v[k] and v[𝓁], respectively, then the induced gap v[k + 1..𝓁 - 1] must be a string from C_{i, j}. This generalises the setting recently investigated in [Day et al., ISAAC 2022], where only gap constraints of the form C_{i, i + 1} are considered, as well as the setting from [Kosche et al., RP 2022], where only gap constraints of the form C_{1, |u|} are considered. We show that subsequence matching under generalised gap constraints is NP-hard, and we complement this general lower bound with a thorough (parameterised) complexity analysis. Moreover, we identify several efficiently solvable subclasses that result from restricting the interval structure induced by the generalised gap constraints.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Formal languages and automata theory
Keywords
  • String algorithms
  • subsequences with gap constraints
  • pattern matching
  • fine-grained complexity
  • conditional lower bounds
  • parameterised complexity

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