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Acta Informatica

, Volume 50, Issue 3, pp 157–173 | Cite as

Avoiding cross-bifix-free binary words

  • Stefano Bilotta
  • Elisabetta Grazzini
  • Elisa Pergola
  • Renzo Pinzani
Original Article

Abstract

In this paper we study the construction and the enumeration of binary words in \(\{0,1\}^*\) having more 1’s than 0’s and avoiding a set of cross-bifix-free patterns. We give a particular succession rule, called jumping and marked succession rule, which describes the growth of such words according to their number of ones. Moreover, the problem of associating a word to a path in the generating tree obtained by the succession rule is solved by introducing an algorithm which constructs all binary words having more 1’s than 0’s and then kills those containing the forbidden patterns. Finally, we give the generating function of such words according to the number of ones.

Keywords

Generate Tree Intrusion Detection Lattice Path Access Control Model Catalan Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Apostolico, A., Atallah, M.: Compact recognizers of episode sequences. Inf. Comput. 174(2), 180–192 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bacchelli, S., Barcucci, E., Grazzini, E., Pergola, E.: Exhaustive generation of combinatorial objects by ECO. Acta Inform. 40(8), 585–602 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bajic, D.: On Construction of Cross-Bifix-Free Kernel Sets, 2nd MCM COST 2100, TD(07) 237. Lisbon, Portugal (2007)Google Scholar
  4. 4.
    Bajic, D., Drajic, D.: Duration of search for a fixed pattern in random data: distribution function and variance. Electron. lett. 31(8), 631–632 (1995)CrossRefGoogle Scholar
  5. 5.
    Bajic, D., Stojanovic, J.: Distributed sequences and search process. In: IEEE International Conference on Communications, ICC2004 Paris, pp. 514–518 (2004)Google Scholar
  6. 6.
    Banderier, C., Bousquet-Mélou, M., Denise, A., Flajolet, P., Gardy, D., Gouyou-Beauchamps, D.: Generating functions for generating trees. Discret. Math. 246, 29–55 (2002)zbMATHCrossRefGoogle Scholar
  7. 7.
    Barcucci, E., Del Lungo, A., Pergola, E., Pinzani, R.: ECO: a methodology for the enumeration of combinatorial objects. J. Differ. Equ. Appl. 5, 435–490 (1999)zbMATHCrossRefGoogle Scholar
  8. 8.
    Bernini, A., Grazzini, E., Pergola, E., Pinzani, R.: A general exhaustive generation algorithm for gray structures. Acta Inform. 44(5), 361–376 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bilotta, S., Merlini, D., Pergola, E., Pinzani, R.: Pattern \(1^{j+1}0^j\) avoiding binary words. Fundam. Inform. 177(1–4), 35–55 (2012)MathSciNetGoogle Scholar
  10. 10.
    Bilotta, S., Pergola, E., Pinzani, R.: A new approach to cross-bifix-free sets. IEEE Trans. Inf. Theory 58(6), 4058–4063 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Corteel, S.: Séries génératrices exponentielles pour les ECO-systèmes signés. In: Proceedings of the 12th International Conference on Formal Power Series and Algebraic Combinatorics, Moscow (2000)Google Scholar
  12. 12.
    Ferrari, L., Pergola, E., Pinzani, R., Rinaldi, S.: Jumping succession rules and their generating functions. Discret. Math. 271, 29–50 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ferrari, L., Pinzani, R.: A linear operator approach to succession rules. Linear Algebra Appl. 348, 231–246 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Flajolet, P., Szpankowski, W., Valle, B.: Hidden word statistics. J. ACM 53(1), 147–183 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guibas, L.J., Odlyzko, M.: Long repetitive patterns in random sequences. Zeitschrift für Wahrscheinlichkeitstheorie 53, 241–262 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Guibas, L.J., Odlyzko, M.: String overlaps, pattern matching, and nontransitive games. J. Comb. Theory Ser. A 30, 183–208 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Harju, T., Nowotka, D.: Counting bordered and primitive words with a fixed weight. Theor. Comput. Sci. 340, 273–279 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kim, K.H., Putcha, M.S., Roush, F.W.: Some combinatorial properties of free semigroups. J. Lond. Math. Soc. 16(2), 397–402 (1977)Google Scholar
  19. 19.
    Kumar, S., Spafford, E.H.: A pattern matching model for misuse intrusion detection. In: Computer Security, pp. 11–21 (1994)Google Scholar
  20. 20.
    Merlini, D., Sprugnoli, R., Verri, M.C.: An Algebra for proper generating tree, In: Algorithms, Trees, Combinatorics and Probabilities, Trends in Mathematics, Mathematics and Computer Science, pp. 127–139 (2000)Google Scholar
  21. 21.
    Nielsen, P.T.: On the expected duration of a search for a fixed pattern in random data. IEEE Trans. Inform. Theory, IT-29, 702–704 (1973)Google Scholar
  22. 22.
    Nielsen, P.T.: A Note on Bifix-free sequences. IEEE Trans. Inform. Theory, IT-29, 704–706 (1973)Google Scholar
  23. 23.
    Rigoutsos, I., Floratos, A., Parida, L., Gao, Y., Platt, D.: The emergence of pattern discovery techniques in computational biology. Metab. Eng. 2, 159–177 (2000)CrossRefGoogle Scholar
  24. 24.
    Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms. Chapman-Hall, London (1995)Google Scholar
  25. 25.
    Waterman, M.: Introduction to Computational Biology. Addison-Wesley, Reading (1995)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stefano Bilotta
    • 1
  • Elisabetta Grazzini
    • 1
  • Elisa Pergola
    • 1
  • Renzo Pinzani
    • 1
  1. 1.Dipartimento di Sistemi e InformaticaUniversità degli Studi di Firenze FirenzeItaly

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