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Back-To-Front Online Lyndon Forest Construction

Authors Golnaz Badkobeh , Maxime Crochemore , Jonas Ellert , Cyril Nicaud

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Author Details

Golnaz Badkobeh
  • Goldsmiths University of London, UK
Maxime Crochemore
  • Univ. Gustave Eiffel, Champs-sur-Marne, France
  • King’s College London, UK
Jonas Ellert
  • Department of Computer Science, Technical University of Dortmund, Germany
Cyril Nicaud
  • LIGM, Univ. Gustave Eiffel, Champs-sur-Marne, France

Funding

  • Ellert, Jonas: Partially supported by the French embassy in Germany (Procope mobility grant, project 0185-DEU-21-016 code 174).

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Golnaz Badkobeh, Maxime Crochemore, Jonas Ellert, and Cyril Nicaud. Back-To-Front Online Lyndon Forest Construction. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CPM.2022.13

Abstract

A Lyndon word is a word that is lexicographically smaller than all of its non-trivial rotations (e.g. ananas is a Lyndon word; banana is not a Lyndon word due to its smaller rotation abanan). The Lyndon forest (or equivalently Lyndon table) identifies maximal Lyndon factors of a word, and is of great combinatoric interest, e.g. when finding maximal repetitions in words. While optimal linear time algorithms for computing the Lyndon forest are known, none of them work in an online manner. We present algorithms that compute the Lyndon forest of a word in a reverse online manner, processing the input word from back to front. We assume a general ordered alphabet, i.e. the only elementary operations on symbols are comparisons of the form less-equal-greater. We start with a naive algorithm and show that, despite its quadratic worst-case behaviour, it already takes expected linear time on words drawn uniformly at random. We then introduce a much more sophisticated algorithm that takes linear time in the worst case. It borrows some ideas from the offline algorithm by Bille et al. (ICALP 2020), combined with new techniques that are necessary for the reverse online setting. While the back-to-front approach for this computation is rather natural (see Franek and Liut, PSC 2019), the steps required to achieve linear time are surprisingly intricate. We envision that our algorithm will be useful for the online computation of maximal repetitions in words.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Combinatorics on words
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Lyndon factorisation
  • Lyndon forest
  • Lyndon table
  • Lyndon array
  • right Lyndon tree
  • Cartesian tree
  • standard factorisation
  • online algorithms

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References

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