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Unveiling the Connection Between the Lyndon Factorization and the Canonical Inverse Lyndon Factorization via a Border Property

Authors Paola Bonizzoni , Clelia De Felice , Brian Riccardi , Rocco Zaccagnino , Rosalba Zizza

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Author Details

Paola Bonizzoni
  • Dip. di Informatica, Sistemistica e Comunicazione, University of Milano-Bicocca, Milan, Italy
Clelia De Felice
  • Dip. di Informatica, University of Salerno, Fisciano, Italy
Brian Riccardi
  • Dip. di Informatica, Sistemistica e Comunicazione, University of Milano-Bicocca, Milan, Italy
Rocco Zaccagnino
  • Dip. di Informatica, University of Salerno, Fisciano, Italy
Rosalba Zizza
  • Dip. di Informatica, University of Salerno, Fisciano, Italy

Funding

  • Bonizzoni, Paola: from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement PANGAIA No. 872539, and from MUR 2022YRB97K, PINC, Pangenome INformatiCs: from Theory to Applications.
  • De Felice, Clelia: from MUR 2022YRB97K, PINC, Pangenome INformatiCs: from Theory to Applications.
  • Riccardi, Brian: from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement PANGAIA No. 872539, and from MUR 2022YRB97K, PINC, Pangenome INformatiCs: from Theory to Applications.
  • Zaccagnino, Rocco: from INdAM-GNCS Project 2023.
  • Zizza, Rosalba: from INdAM-GNCS Project 2023.

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Paola Bonizzoni, Clelia De Felice, Brian Riccardi, Rocco Zaccagnino, and Rosalba Zizza. Unveiling the Connection Between the Lyndon Factorization and the Canonical Inverse Lyndon Factorization via a Border Property. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.31

Abstract

The notion of Lyndon word and Lyndon factorization has shown to have unexpected applications in theory as well as in developing novel algorithms on words. A counterpart to these notions are those of inverse Lyndon word and inverse Lyndon factorization. Differently from the Lyndon words, the inverse Lyndon words may be bordered. The relationship between the two factorizations is related to the inverse lexicographic ordering, and has only been recently explored. More precisely, a main open question is how to get an inverse Lyndon factorization from a classical Lyndon factorization under the inverse lexicographic ordering, named CFL_in. In this paper we reveal a strong connection between these two factorizations where the border plays a relevant role. More precisely, we show two main results. We say that a factorization has the border property if a nonempty border of a factor cannot be a prefix of the next factor. First we show that there exists a unique inverse Lyndon factorization having the border property. Then we show that this unique factorization with the border property is the so-called canonical inverse Lyndon factorization, named ICFL. By showing that ICFL is obtained by compacting factors of the Lyndon factorization over the inverse lexicographic ordering, we provide a linear time algorithm for computing ICFL from CFL_in.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Lyndon words
  • Lyndon factorization
  • Combinatorial algorithms on words

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