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Periodicity and Unbordered Words

A Proof of Duval’s Conjecture

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STACS 2004 (STACS 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2996))

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Abstract

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper.

A word is bordered, if it has a proper prefix that is also a suffix of that word. Consider a finite word w of length n. Let μ(w) denote the maximum length of its unbordered factors, and let \(\partial(w)\) denote the period of w. Clearly, \(\mu(w) \leq \partial(w)\).

We establish that \(\mu(w) = \partial(w)\), if w has an unbordered prefix of length μ(w) and n ≥ 2μ(w). This bound is tight and solves a 21 year old conjecture by Duval. It follows from this result that, in general, n ≥ 3μ(w) implies \(\mu(w) = \partial(w)\) which gives an improved bound for the question asked by Ehrenfeucht and Silberger in 1979.

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References

  1. Assous, R., Pouzet, M.: Une caractérisation des mots périodiques. Discrete Math. 25(1), 1–5 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berstel, J., Perrin, D.: Theory of codes. Pure and Applied Mathematics, vol. 117. Academic Press Inc., Orlando (1985)

    MATH  Google Scholar 

  3. Boyer, R.S., Moore, J.S.: A fast string searching algorithm. Commun. ACM 20(10), 762–772 (1977)

    Article  Google Scholar 

  4. Breslauer, D., Jiang, T., Jiang, Z.: Rotations of periodic strings and short superstrings. J. Algorithms 24(2) (1997)

    Google Scholar 

  5. Bylanski, P., Ingram, D.G.W.: Digital transmission systems. Telecommunications Series, vol. 4. IEEE, Los Alamitos (1980)

    Google Scholar 

  6. Crochemore, M., Mignosi, F., Restivo, A., Salemi, S.: Text compression using antidictionaries. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 261–270. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  7. Crochemore, M., Perrin, D.: Two-way string-matching. J. ACM 38(3), 651–675 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Duval, J.-P.: Relationship between the period of a finite word and the length of its unbordered segments. Discrete Math. 40(1), 31–44 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duval, J.-P., Harju, T., Nowotka, D.: Unbordered factors and Lyndon words (submitted)

    Google Scholar 

  10. Ehrenfeucht, A., Silberger, D.M.: Periodicity and unbordered segments of words. Discrete Math. 26(2), 101–109 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. Harju, T., Nowotka, D.: About Duval’s conjecture. In: Ésik, Z., Fülöp, Z. (eds.) DLT 2003. LNCS, vol. 2710, pp. 316–324. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  12. Harju, T., Nowotka, D.: Periodicity and unbordered words. TUCS Tech. Rep. 523, Turku Centre of Computer Science, Finland (April 2003)

    Google Scholar 

  13. Holub, S.: A proof of Duval’s conjecture. In: Harju, T., Karhumäki, J. (eds.) Optimization Techniques 1974. Turku Centre of Computer Science, vol. 27, pp. 398–399. TUCS General Publications (2003)

    Google Scholar 

  14. Holub, S.: Unbordered words and lexicographic orderings. personal communication (July 2003)

    Google Scholar 

  15. Knuth, D.E., Morris, J.H., Pratt, V.R.: Fast Pattern matching in strings. SIAM J. Comput. 6(2), 323–350 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  17. Margaritis, D., Skiena, S.: Reconstructing strings from substrings in rounds. In: 36th Annual Symposium on Foundations of Computer Science (FOCS), Milwaukee, WI, pp. 613–620. IEEE Computer Society, Los Alamitos (1995)

    Google Scholar 

  18. Mignosi, F., Zamboni, L.Q.: A note on a conjecture of Duval and Sturmian words. Theor. Inform. Appl. 36(1), 1–3 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Trans. Information Theory 23(3), 337–343 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Turku Centre for Computer Science (TUCS) Department of Mathematics, University of Turku, Finland

    Tero Harju & Dirk Nowotka

Authors
  1. Tero Harju
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  2. Dirk Nowotka
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Editor information

Editors and Affiliations

  1. Universität Stuttgart, FMI, Germany

    Volker Diekert

  2. LIAFA and the University of Paris 7, Denis Diderot,  

    Michel Habib

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© 2004 Springer-Verlag Berlin Heidelberg

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Harju, T., Nowotka, D. (2004). Periodicity and Unbordered Words. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_26

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  • DOI: https://doi.org/10.1007/978-3-540-24749-4_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21236-2

  • Online ISBN: 978-3-540-24749-4

  • eBook Packages: Springer Book Archive

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