Back-To-Front Online Lyndon Forest Construction

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Title of host publication33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022
EditorsHideo Bannai, Jan Holub
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772341
Publication statusPublished - 1 Jun 2022
Event33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022 - Prague, Czech Republic
Duration: 27 Jun 202229 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022
Country/TerritoryCzech Republic


  • Cartesian tree
  • Lyndon array
  • Lyndon factorisation
  • Lyndon forest
  • Lyndon table
  • online algorithms
  • right Lyndon tree
  • standard factorisation


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