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Efficient Computation of 2-Covers of a String

Authors Jakub Radoszewski , Juliusz Straszyński

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Author Details

Jakub Radoszewski
  • Institute of Informatics, University of Warsaw, Poland
  • Samsung R&D Poland, Warsaw, Poland
Juliusz Straszyński
  • Institute of Informatics, University of Warsaw, Poland

Funding

Supported by the "Algorithms for text processing with errors and uncertainties" project carried out within the HOMING program of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund, project no. POIR.04.04.00-00-24BA/16, and by the Polish National Science Center, grant no. 2018/31/D/ST6/03991.

Acknowledgements

The authors thank Patryk Czajka for helpful discussions on the initial version of the algorithm.

Cite AsGet BibTex

Jakub Radoszewski and Juliusz Straszyński. Efficient Computation of 2-Covers of a String. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.77

Abstract

Quasiperiodicity is a generalization of periodicity that has been researched for almost 30 years. The notion of cover is the classic variant of quasiperiodicity. A cover of a text T is a string whose occurrences in T cover all positions of T. There are several algorithms computing covers of a text in linear time. In this paper we consider a natural extension of cover. For a text T, we call a pair of strings a 2-cover if they have the same length and their occurrences cover the text T. We give an algorithm that computes all 2-covers of a string of length n in 𝒪(n log n log log n + output) expected time or 𝒪(n log n log² log n / log log log n + output) worst-case time, where output is the size of output. If (X,Y) is a 2-cover of T, then either X is a prefix and Y is a suffix of T, in which case we call (X,Y) a ps-cover, or one of X, Y is a border (that is, both a prefix and a suffix) of T, and then we call (X,Y) a b-cover. A string of length n has up to n ps-covers; we show an algorithm that computes all of them in 𝒪(n log log n) expected time or 𝒪(n log² log n / log log log n) worst-case time. A string of length n can have Θ(n²) non-trivial b-covers; our algorithm can report one b-cover per length (if it exists) or all shortest b-covers in 𝒪(n log n log log n) expected time or 𝒪(n log n log² log n / log log log n) worst-case time. All our algorithms use linear space. The problem in scope can be generalized to λ > 2 equal-length strings, resulting in the notion of λ-cover. Cole et al. (2005) showed that the λ-cover problem is NP-complete. Our algorithms generalize to λ-covers, with (the first component of) the algorithm’s complexity multiplied by n^{λ-2}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • quasiperiodicity
  • cover of a string
  • 2-cover
  • lambda-cover

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