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Fun Slot Machines and Transformations of Words Avoiding Factors

Authors Marcella Anselmo, Manuela Flores, Maria Madonia

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Formal languages and automata theory
Keywords
  • Isometric words
  • Words avoiding factors
  • Index of a word
  • Overlap
  • Lee distance

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Abstract

Fun Slot Machines are a variant of the classical ones. Pulling a lever, the player generates a sequence of symbols which are placed on the reels. The machine pays when a given pattern appears in the sequence. The variant consists in trying to transform a losing sequence of symbols in another one, in such a way that the winning pattern does not appear in any intermediate step. The choice of the winning pattern can be crucial; there are "good" and "bad" sequences. The game results in a combinatorial problem on transformations of words avoiding a given pattern as a factor. We investigate "good" and "bad" sequences on a k-ary alphabet and the pairs of words that witness that a word is "bad". A main result is an algorithm to decide whether a word is "bad" or not and to provide a pair of witnesses of minimal length when the word is "bad". It runs in O(n) time with a preprocessing of O(n) time and space to construct an enhanced suffix tree of the word.

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Marcella Anselmo, Manuela Flores, and Maria Madonia. Fun Slot Machines and Transformations of Words Avoiding Factors. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.FUN.2022.4

Author Details

Marcella Anselmo
  • Dipartimento di Informatica, University of Salerno, Fisciano (SA), Italy
Manuela Flores
  • Dipartimento di Informatica, University of Salerno, Fisciano (SA), Italy
Maria Madonia
  • Dipartimento di Matematica e Informatica, University of Catania, Italy

Funding

Partially Supported by INdAM-GNCS Project 2021, FARB Project ORSA218107 of University of Salerno and TEAMS Project of University of Catania.

References

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  2. Marcella Anselmo, Manuela Flores, and Maria Madonia. Quaternary n-cubes and isometric words. In Thierry Lecroq and Svetlana Puzynina, editors, Combinatorics on Words, pages 27-39, Cham, 2021. Springer International Publishing. Google Scholar
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