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Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform

Authors Gabriele Fici , Estéban Gabory

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ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
Keywords
  • Burrows-Wheeler Transform
  • Generalized de Bruijn Word
  • Generalized de Bruijn Graph
  • Circulant Matrix
  • Invertible Necklace
  • Sandpile Group
  • Reutenauer Group

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Abstract

We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles in the generalized de Bruijn graphs, introduced in the early '80s in the context of network design. When the size of the alphabet is a prime p, we define invertible necklaces as those whose BWT-matrix is non-singular. We show that invertible necklaces of length n correspond to normal bases of the finite field 𝔽_{pⁿ}, and that they form an Abelian group isomorphic to the Reutenauer group RG_pⁿ. Using known results in abstract algebra, we can make a bridge between generalized de Bruijn words and invertible necklaces. In particular, we highlight a correspondence between binary de Bruijn words of order d+1, binary necklaces of length 2^{d} having an odd number of 1’s, invertible BWT matrices of size 2^{d}× 2^{d}, and normal bases of the finite field 𝔽_{2^{2^{d}}}.

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Gabriele Fici and Estéban Gabory. Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.48

Author Details

Gabriele Fici
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy
Estéban Gabory
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy

Funding

  • Fici, Gabriele: Supported by MIUR project PRIN 2022 APML – 20229BCXNW.
  • Gabory, Estéban: Supported by MIUR project PRIN 2022 APML – 20229BCXNW.

References

  1. Yu Hin Au. Generalized de Bruijn words for primitive words and powers. Discret. Math., 338(12):2320-2331, 2015. URL: https://doi.org/10.1016/J.DISC.2015.05.025.
  2. D. J. Aulicino and Morris Goldfeld. A new relation between primitive roots and permutations. The American Mathematical Monthly, 76(6):664-666, 1969. URL: https://doi.org/10.1080/00029890.1969.12000297.
  3. Jean Berstel, Aaron Lauve, Christophe Reutenauer, and Franco Saliola. Combinatorics on Words: Christoffel Words and Repetition in Words, volume 27 of CRM monograph series. American Mathematical Society, 2008. Google Scholar
  4. Lenore Blum, Manuel Blum, and Michael Shub. Comparison of Two Pseudo-Random Number Generators. In David Chaum, Ronald L. Rivest, and Alan T. Sherman, editors, Advances in Cryptology, pages 61-78, Boston, MA, 1983. Springer US. Google Scholar
  5. Swee Hong Chan, Henk D. L. Hollmann, and Dmitrii V. Pasechnik. Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields. In Jaroslav Nešetřil and Marco Pellegrini, editors, The Seventh European Conference on Combinatorics, Graph Theory and Applications, pages 97-104, Pisa, 2013. Scuola Normale Superiore. Google Scholar
  6. Swee Hong Chan, Henk D.L. Hollmann, and Dmitrii V. Pasechnik. Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields. Journal of Algebra, 421:268-295, 2015. Special issue in memory of Ákos Seress. URL: https://doi.org/10.1016/j.jalgebra.2014.08.029.
  7. Yaotsu Chang, T. K. Truong, and I. S. Reed. Normal bases over GF(q). Journal of Algebra, 241:89-101, 2001. Google Scholar
  8. Maxime Crochemore, Jacques Désarménien, and Dominique Perrin. A note on the burrows - wheeler transformation. Theor. Comput. Sci., 332(1-3):567-572, 2005. URL: https://doi.org/10.1016/J.TCS.2004.11.014.
  9. Maxime Crochemore, Christophe Hancart, and Thierry Lecroq. Algorithms on strings. Cambridge University Press, 2007. Google Scholar
  10. Ding-Zhu Du and Frank K. Hwang. Generalized de Bruijn digraphs. Networks, 18(1):27-38, 1988. URL: https://doi.org/10.1002/NET.3230180105.
  11. S. V. Duzhin and D. V. Pasechnik. Groups acting on necklaces and sandpile groups. Journal of Mathematical Sciences, 200(6):690-697, 2014. URL: https://doi.org/10.1007/s10958-014-1960-6.
  12. Gabriele Fici, Sabrina Mantaci, Antonio Restivo, Giuseppe Romana, Giovanna Rosone, and Marinella Sciortino. BWT and Combinatorics on Words. In Paolo Ferragina, Travis Gagie, and Gonzalo Navarro, editors, The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini’s 60th Birthday, volume 131 of OASIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. Google Scholar
  13. K. Hensel. Ueber die darstellung der zahlen eines gattungsbereiches für einen beliebigen primdivisor. Journal für die reine und angewandte Mathematik, 1888(103):230-237, 1888. URL: https://doi.org/10.1515/crll.1888.103.230.
  14. Peter M. Higgins. Burrows-Wheeler transformations and de Bruijn words. Theor. Comput. Sci., 457:128-136, 2012. URL: https://doi.org/10.1016/J.TCS.2012.07.019.
  15. Trevor Hyde. Normal elements in finite fields, 2018. URL: https://arxiv.org/abs/1809.02155.
  16. Makoto Imase and Masaki Itoh. Design to minimize diameter on building-block network. IEEE Trans. Computers, 30(6):439-442, 1981. URL: https://doi.org/10.1109/TC.1981.1675809.
  17. G. Kirchhoff. On the solution of the equations obtained from the investigation of the linear distribution of galvanic currents. IRE Transactions on Circuit Theory, 5(1):4-7, 1958. URL: https://doi.org/10.1109/TCT.1958.1086426.
  18. Ben Langmead and Steven L. Salzberg. Fast gapped-read alignment with Bowtie 2. Nat. Methods, 9(4):357-359, 2012. Google Scholar
  19. Ben Langmead, Cole Trapnell, Mihai Pop, and Steven L Salzberg. Ultrafast and memory-efficient alignment of short DNA sequences to the human genome. Genome Biol., 10(3):R25, 2009. Google Scholar
  20. Lionel Levine. Sandpile groups and spanning trees of directed line graphs. J. Comb. Theory A, 118(2):350-364, 2011. URL: https://doi.org/10.1016/J.JCTA.2010.04.001.
  21. Heng Li and Richard Durbin. Fast and accurate long-read alignment with Burrows-Wheeler transform. Bioinformatics, 26(5):589-595, 2010. URL: https://doi.org/10.1093/BIOINFORMATICS/BTP698.
  22. Xueliang Li and Fuji Zhang. On the numbers of spanning trees and Eulerian tours in generalized de Bruijn graphs. Discret. Math., 94(3):189-197, 1991. URL: https://doi.org/10.1016/0012-365X(91)90024-V.
  23. Rudolf Lidl and Harald Niederreiter. Finite Fields. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 1996. Google Scholar
  24. Zsuzsanna Lipták and Luca Parmigiani. A BWT-Based Algorithm for Random de Bruijn Sequence Construction. In José A. Soto and Andreas Wiese, editors, LATIN 2024: Theoretical Informatics - 16th Latin American Symposium, Puerto Varas, Chile, March 18-22, 2024, Proceedings, Part I, volume 14578 of Lecture Notes in Computer Science, pages 130-145. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-55598-5_9.
  25. M. Lothaire. Combinatorics on Words. Advanced Book Program. Addison-Wesley, Advanced Book Program, World Science Division, 1983. Google Scholar
  26. Sabrina Mantaci, Antonio Restivo, Giovanna Rosone, Marinella Sciortino, and Luca Versari. Measuring the clustering effect of BWT via RLE. Theor. Comput. Sci., 698:79-87, 2017. URL: https://doi.org/10.1016/J.TCS.2017.07.015.
  27. Sabrina Mantaci, Antonio Restivo, and Marinella Sciortino. Burrows-Wheeler transform and Sturmian words. Inf. Process. Lett., 86(5):241-246, 2003. URL: https://doi.org/10.1016/S0020-0190(02)00512-4.
  28. Abhinav Nellore and Rachel A. Ward. Arbitrary-Length Analogs to de Bruijn Sequences. In Hideo Bannai and Jan Holub, editors, 33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022, June 27-29, 2022, Prague, Czech Republic, volume 223 of LIPIcs, pages 9:1-9:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.CPM.2022.9.
  29. S.M. Reddy, D.K. Pradhan, and J.G. Kuhl. Directed graphs with minimum diameter and maximal connectivity. Technical report, School of Engineering, Oakland University, July 1980. Google Scholar
  30. Christophe Reutenauer and Jeffrey O. Shallit. Christoffel Matrices and Sturmian Determinants. CoRR, abs/2409.09824, 2024. URL: https://doi.org/10.48550/arXiv.2409.09824.
  31. Giovanna Rosone and Marinella Sciortino. The Burrows-Wheeler Transform between Data Compression and Combinatorics on Words. In Paola Bonizzoni, Vasco Brattka, and Benedikt Löwe, editors, The Nature of Computation. Logic, Algorithms, Applications - 9th Conference on Computability in Europe, CiE 2013, Milan, Italy, July 1-5, 2013. Proceedings, volume 7921 of Lecture Notes in Computer Science, pages 353-364. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-39053-1_42.
  32. N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. Available electronically at URL: http://oeis.org.
  33. Richard P. Stanley. Smith normal form in combinatorics. J. Comb. Theory A, 144:476-495, 2016. URL: https://doi.org/10.1016/J.JCTA.2016.06.013.
  34. T. van Aardenne-Ehrenfest and N.G. de Bruijn. Circuits and Trees in Oriented Linear Graphs, pages 149-163. Birkhäuser Boston, Boston, MA, 1987. URL: https://doi.org/10.1007/978-0-8176-4842-8_12.
  35. Luca Q. Zamboni. On Christoffel words and their lexicographic array. CoRR, abs/2409.07974, 2024. URL: https://doi.org/10.48550/arXiv.2409.07974.
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