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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.

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