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Totally Unclustered BWT Images of Any Length over Non-Binary Alphabets

Authors Gabriele Fici , Estéban Gabory , Giuseppe Romana , Marinella Sciortino

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LIPIcs.CPM.2026.13.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Data structures design and analysis
Keywords
  • Burrows-Wheeler Transform
  • BWT-runs
  • Repetitiveness Measure
  • Clustering Effect
  • Generalized de Bruijn Words

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Abstract

We prove that for every integer n > 0 and for every alphabet Σ_k of size k ≥ 3, there exist words of length n whose Burrows-Wheeler Transform (BWT) is totally unclustered, i.e., it consists of exactly n runs with no two consecutive equal symbols. These words represent the worst-case behavior of the clustering effect of the BWT. We also establish a lower bound on their number. This contrasts with the binary case, where the existence of infinitely many totally unclustered BWT images is still an open problem, related to Artin’s conjecture on primitive roots.

Cite As Get BibTex

Gabriele Fici, Estéban Gabory, Giuseppe Romana, and Marinella Sciortino. Totally Unclustered BWT Images of Any Length over Non-Binary Alphabets. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.13

Author Details

Gabriele Fici
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy
Estéban Gabory
  • Institute of Computer Science, University of Wrocław, Poland
Giuseppe Romana
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy
Marinella Sciortino
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy

Funding

  • Fici, Gabriele: Supported by MIUR project PRIN 2022 APML – 20229BCXNW.
  • Gabory, Estéban: Partially supported by MIUR project PRIN 2022 APML – 20229BCXNW. Partially supported by the Polish National Science Centre grant number 2023/51/B/ST6/01505.
  • Romana, Giuseppe: Supported by the project ACoMPA – Algorithmic and Combinatorial Methods for Pangenome Analysis (CUP B73C24001050001) funded by the NextGeneration EU programme PNRR MUR M4 C2 Inv. 1.5 - Project ECS00000017 Tuscany Health Ecosystem (Spoke 6), CUP Master B63C22000680007, and by the INdAM - GNCS Project CUP_E53C25002010001.
  • Sciortino, Marinella: Partially supported by the project ACoMPA – Algorithmic and Combinatorial Methods for Pangenome Analysis (CUP B73C24001050001) funded by the NextGeneration EU programme PNRR MUR M4 C2 Inv. 1.5 - Project ECS00000017 Tuscany Health Ecosystem (Spoke 6), CUP Master B63C22000680007, and by the INdAM - GNCS Project CUP_E53C25002010001.

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