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Linear Time Construction of Cover Suffix Tree and Applications

Author Jakub Radoszewski

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Author Details

Jakub Radoszewski
  • University of Warsaw, Poland
  • Samsung R&D, Warsaw, Poland

Funding

  • Radoszewski, Jakub: Supported by the Polish National Science Center, grant no. 2022/46/E/ST6/00463.

Acknowledgements

The author thanks the anonymous reviewers for reading the manuscript carefully and providing several useful suggestions.

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Jakub Radoszewski. Linear Time Construction of Cover Suffix Tree and Applications. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.89

Abstract

The Cover Suffix Tree (CST) of a string T is the suffix tree of T with additional explicit nodes corresponding to halves of square substrings of T. In the CST an explicit node corresponding to a substring C of T is annotated with two numbers: the number of non-overlapping consecutive occurrences of C and the total number of positions in T that are covered by occurrences of C in T. Kociumaka et al. (Algorithmica, 2015) have shown how to compute the CST of a length-n string in 𝒪(n log n) time. We give an algorithm that computes the same data structure in 𝒪(n) time assuming that T is over an integer alphabet and discuss its implications. A string C is a cover of text T if occurrences of C in T cover all positions of T; C is a seed of T if occurrences and overhangs (i.e., prefix-suffix occurrences) of C in T cover all positions of T. An α-partial cover (α-partial seed) of text T is a string C whose occurrences in T (occurrences and overhangs in T, respectively) cover at least α positions of T. Kociumaka et al. (Algorithmica, 2015; Theor. Comput. Sci., 2018) have shown that knowing the CST of a length-n string T, one can compute a linear-sized representation of all seeds of T as well as all shortest α-partial covers and seeds in T for a given α in 𝒪(n) time. Thus our result implies linear-time algorithms computing these notions of quasiperiodicity. The resulting algorithm computing seeds is substantially different from the previous one (Kociumaka et al., SODA 2012, ACM Trans. Algorithms, 2020); in particular, it is non-recursive. Kociumaka et al. (Algorithmica, 2015) proposed an 𝒪(n log n)-time algorithm for computing a shortest α-partial cover for each α = 1,…,n; we improve this complexity to 𝒪(n). Our results are based on a new combinatorial characterization of consecutive overlapping occurrences of a substring S of T in terms of the set of runs (see Kolpakov and Kucherov, FOCS 1999) in T. This new insight also leads to an 𝒪(n)-sized index for reporting overlapping consecutive occurrences of a given pattern P of length m in the optimal 𝒪(m+output) time, where output is the number of occurrences reported. In comparison, a general index for reporting bounded-gap consecutive occurrences of Navarro and Thankachan (Theor. Comput. Sci., 2016) uses 𝒪(n log n) space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • cover (quasiperiod)
  • seed
  • suffix tree
  • run (maximal repetition)

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