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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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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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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.

Cite As Get BibTex

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

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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