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On Extensions of Maximal Repeats in Compressed Strings

Author Julian Pape-Lange

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ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Mathematics of computing → Combinatoric problems
Keywords
  • Maximal repeats
  • Extensions of maximal repeats
  • Combinatorics on compressed strings
  • LZ77
  • Burrows-Wheeler transform
  • Burrows-Wheeler transform conjecture
  • Compact suffix automata
  • CDAWGs

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Abstract

This paper provides upper bounds for several subsets of maximal repeats and maximal pairs in compressed strings and also presents a formerly unknown relationship between maximal pairs and the run-length Burrows-Wheeler transform.
This relationship is used to obtain a different proof for the Burrows-Wheeler conjecture which has recently been proven by Kempa and Kociumaka in "Resolution of the Burrows-Wheeler Transform Conjecture".
More formally, this paper proves that the run-length Burrows-Wheeler transform of a string S with z_S LZ77-factors has at most 73(log₂ |S|)(z_S+2)² runs, and if S does not contain q-th powers, the number of arcs in the compacted directed acyclic word graph of S is bounded from above by 18q(1+log_q |S|)(z_S+2)².

Cite As Get BibTex

Julian Pape-Lange. On Extensions of Maximal Repeats in Compressed Strings. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CPM.2020.27

Author Details

Julian Pape-Lange
  • Technische Universität Chemnitz, Straße der Nationen 62, 09111 Chemnitz, Germany

Acknowledgements

Fabio Cunial suggested that my previous work might be extendable from counting maximal repeats to counting extensions of maximal repeats. He also pointed out that such a result would be more interesting since it is more closely linked to the size of the compacted directed acyclic word graph. Nicola Prezza noted that my previous work also resulted in a non-trivial upper bound for the number of runs in the run-length Burrows-Wheeler transform and that a more careful investigation of the extensions of maximal repeats might result in a better bound for the Burrows-Wheeler conjecture which was unsolved at that time. I also thank Djamal Belazzougui for notifying me of the "Resolution of the Burrows-Wheeler Conjecture" by Kempa and Kociumaka.

References

  1. Djamal Belazzougui, Fabio Cunial, Travis Gagie, Nicola Prezza, and Mathieu Raffinot. Composite repetition-aware data structures. In Ferdinando Cicalese, Ely Porat, and Ugo Vaccaro, editors, Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Ischia Island, Italy, June 29 - July 1, 2015, Proceedings, volume 9133 of Lecture Notes in Computer Science, pages 26-39. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-19929-0_3.
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  7. I. Furuya, T. Takagi, Y. Nakashima, S. Inenaga, H. Bannai, and T. Kida. Mr-repair: Grammar compression based on maximal repeats. In 2019 Data Compression Conference (DCC), pages 508-517, March 2019. URL: https://doi.org/10.1109/DCC.2019.00059.
  8. Dominik Kempa and Tomasz Kociumaka. Resolution of the burrows-wheeler transform conjecture. CoRR, abs/1910.10631, 2019. URL: http://arxiv.org/abs/1910.10631.
  9. Julian Pape-Lange. On Maximal Repeats in Compressed Strings. In Nadia Pisanti and Solon P. Pissis, editors, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019), volume 128 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:13, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CPM.2019.18.
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