Lyndon Words, the Three Squares Lemma, and Primitive Squares

Hideo Bannai; Takuya Mieno; Yuto Nakashima

doi:10.1007/978-3-030-59212-7_19

Lyndon Words, the Three Squares Lemma, and Primitive Squares

Hideo Bannai, Takuya Mieno, Yuto Nakashima

Mathematical Informatics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

We revisit the so-called “Three Squares Lemma” by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a common prefix. We also give an improved upper bound of on the maximum number of (occurrences of) primitively rooted squares in a string of length n, also using arguments based on Lyndon words. To the best of our knowledge, the only known upper bound was, where is the golden ratio, reported by Fraenkel and Simpson [TCS 1999] obtained via the Three Squares Lemma.

Original language	English
Title of host publication	String Processing and Information Retrieval - 27th International Symposium, SPIRE 2020, Proceedings
Editors	Christina Boucher, Sharma V. Thankachan
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	265-273
Number of pages	9
ISBN (Print)	9783030592110
DOIs	https://doi.org/10.1007/978-3-030-59212-7_19
Publication status	Published - 2020
Event	27th International Symposium on String Processing and Information Retrieval, SPIRE 2020 - Orlando, United States Duration: Oct 13 2020 → Oct 15 2020

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12303 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	27th International Symposium on String Processing and Information Retrieval, SPIRE 2020
Country/Territory	United States
City	Orlando
Period	10/13/20 → 10/15/20

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/978-3-030-59212-7_19

Cite this

Bannai, H., Mieno, T., & Nakashima, Y. (2020). Lyndon Words, the Three Squares Lemma, and Primitive Squares. In C. Boucher, & S. V. Thankachan (Eds.), String Processing and Information Retrieval - 27th International Symposium, SPIRE 2020, Proceedings (pp. 265-273). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12303 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-59212-7_19

Lyndon Words, the Three Squares Lemma, and Primitive Squares. / Bannai, Hideo; Mieno, Takuya; Nakashima, Yuto.

String Processing and Information Retrieval - 27th International Symposium, SPIRE 2020, Proceedings. ed. / Christina Boucher; Sharma V. Thankachan. Springer Science and Business Media Deutschland GmbH, 2020. p. 265-273 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12303 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Bannai, H, Mieno, T & Nakashima, Y 2020, Lyndon Words, the Three Squares Lemma, and Primitive Squares. in C Boucher & SV Thankachan (eds), String Processing and Information Retrieval - 27th International Symposium, SPIRE 2020, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12303 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 265-273, 27th International Symposium on String Processing and Information Retrieval, SPIRE 2020, Orlando, United States, 10/13/20. https://doi.org/10.1007/978-3-030-59212-7_19

Bannai H, Mieno T, Nakashima Y. Lyndon Words, the Three Squares Lemma, and Primitive Squares. In Boucher C, Thankachan SV, editors, String Processing and Information Retrieval - 27th International Symposium, SPIRE 2020, Proceedings. Springer Science and Business Media Deutschland GmbH. 2020. p. 265-273. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-030-59212-7_19

Bannai, Hideo ; Mieno, Takuya ; Nakashima, Yuto. / Lyndon Words, the Three Squares Lemma, and Primitive Squares. String Processing and Information Retrieval - 27th International Symposium, SPIRE 2020, Proceedings. editor / Christina Boucher ; Sharma V. Thankachan. Springer Science and Business Media Deutschland GmbH, 2020. pp. 265-273 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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N2 - We revisit the so-called “Three Squares Lemma” by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a common prefix. We also give an improved upper bound of on the maximum number of (occurrences of) primitively rooted squares in a string of length n, also using arguments based on Lyndon words. To the best of our knowledge, the only known upper bound was, where is the golden ratio, reported by Fraenkel and Simpson [TCS 1999] obtained via the Three Squares Lemma.

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Lyndon Words, the Three Squares Lemma, and Primitive Squares

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