Skip to main content
SpringerLink
Log in

On maximal repetitions in words

  • Conference paper
  • First Online:
Fundamentals of Computation Theory (FCT 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1684))

Included in the following conference series:

Abstract

A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w.

We first study maximal repetitions in Fibonacci words — we count their exact number, and estimate the sum of their exponents. These quantities turn out to be linearly-bounded in the length of the word. We then prove that the maximal number of maximal repetitions in general words (on arbitrary alphabet) of length n is linearly-bounded in n, and we mention some applications and consequences of this result.

Article

The work has been done during the first author’s visit of LORIA/INRIA-Lorraine supported by a grant from the French Ministry of Public Education and Research. The first author has been also in part supported by the Russian Foundation of Fundamental Research, under grant 96-01-01068, and by the Russian Federal Programme “Integration”, under grant 473. The work has been done within a joint project of the French-Russian A.M.Liapunov Institut of Applied Mathematics and Informatics at Moscow University

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Apostolico and F.P. Preparata. Optimal off-line detection of repetitions in a string. Theoretical Computer Science, 22(3):297–315, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Crochemore and W. Rytter. Text algorithms. Oxford University Press, 1994.

    Google Scholar 

  3. M. Crochemore and W. Rytter. Squares, cubes, and time-space efficient string searching. Algorithmica, 13:405–425, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Crochemore. An optimal algorithm for computing the repetitions in a word. Information Processing Letters, 12:244–250, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Crochemore. Recherche linèaire d’un carrè dans un mot. Comptes Rendus Acad. Sci. Paris Sèr. I Math., 296:781–784, 1983.

    MATH  MathSciNet  Google Scholar 

  6. J.D. Currie and R.O. Shelton. Cantor sets and Dejean’s conjecture. Journal of Automata, Languages and Combinatorics, 1(2):113–128, 1996.

    MATH  MathSciNet  Google Scholar 

  7. F. Dejean. Sur un thèorème de Thue. J. Combinatorial Th. (A), 13:90–99, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  8. A.S. Fraenkel and J. Simpson. How many squares must a binary sequence contain? Electronic Journal of Combinatorics, 2(R2):9pp, 1995. http://www.combinatorics.org/Journal/journalhome.html.

    Google Scholar 

  9. A.S. Fraenkel and J. Simpson. How many squares can a string contain? J. Combinatorial Theory (Ser. A), 82:112–120, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.S. Fraenkel and J. Simpson. The exact number of squares in Fibonacci words. Theoretical Computer Science, 218(1):83–94, 1999.

    Article  MathSciNet  Google Scholar 

  11. C.S. Iliopoulos, D. Moore, and W.F. Smyth. A characterization of the squares in a Fibonacci string. Theoretical Computer Science, 172:281–291, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Justin and G. Pirillo. Fractional powers in Sturmian words. Technical Report LIAFA 99/01, Laboratoire d’Informatique Algorithmique: Fondements et Applications (LIAFA), 1999.

    Google Scholar 

  13. R. Kolpakov and G. Kucherov. Maximal repetitions in words or how to find all squares in linear time. Rapport Interne LORIA 98-R-227, Laboratoire Lorrain de Recherche en Informatique et ses Applications, 1998. available from URL: http://www.loria.fr/~kucherov/res_activ.html.

  14. S. R. Kosaraju. Computation of squares in string. In M. Crochemore and D. Gusfield, editors, Proceedings of the 5th Annual Symposium on Combinatorial Pattern Matching, number 807 in Lecture Notes in Computer Science, pages 146–150. Springer Verlag, 1994.

    Google Scholar 

  15. M. Lothaire. Combinatorics on Words, volume 17 of Encyclopedia of Mathematics and Its Applications. Addison Wesley, 1983.

    Google Scholar 

  16. M. G. Main. Detecting leftmost maximal periodicities. Discrete Applied Mathematics, 25:145–153, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  17. M.G. Main and R.J. Lorentz. An O(n log n) algorithm for finding all repetitions in a string. Journal of Algorithms, 5(3):422–432, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  18. M.G. Main and R.J. Lorentz. Linear time recognition of square free strings. In A. Apostolico and Z. Galil, editors, Combinatorial Algorithms on Words, volume 12 of NATO Advanced Science Institutes, Series F, pages 272–278. Springer Verlag, 1985.

    Google Scholar 

  19. F. Mignosi and G. Pirillo. Repetitions in the Fibonacci infinite word. RAIRO Theoretical Informatics and Applications, 26(3):199–204, 1992.

    MATH  MathSciNet  Google Scholar 

  20. F. Mignosi, A. Restivo, and S. Salemi. A periodicity theorem on words and applications. In Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 969 of Lecture Notes in Computer Science, pages 337–348. Springer Verlag, 1995.

    Google Scholar 

  21. P. Sèèbold. Propriètès combinatoires des mots infinis engendrès par certains morphismes. Rapport 85-16, LITP, Paris, 1985.

    Google Scholar 

  22. J. Stoye and D. Gusfield. Simple and flexible detection of contiguous repeats using a suffix tree. In M. Farach-Colton, editor, Proceedings of the 9th Annual Symposium on Combinatorial Pattern Matching, number 1448 in Lecture Notes in Computer Science, pages 140–152. Springer Verlag, 1998.

    Chapter  Google Scholar 

  23. J. Stoye and D. Gusfield. Linear time algorithms for finding and representing all the tandem repeats in a string. Technical Report CSE-98-4, Computer Science Department, University of California, Davis, 1998.

    Google Scholar 

  24. A.O. Slisenko. Detection of periodicities and string matching in real time. Journal of Soviet Mathematics, 22:1316–1386, 1983.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. French-Russian Institute for Informatics and Applied Mathematics, Moscow University, 119899, Moscow, Russia

    Roman Kolpakov

  2. LORIA/INRIA-Lorraine, 615, rue du Jardin Botanique, B.P. 101, 54602, Villers-lès-Nancy, France

    Gregory Kucherov

Authors
  1. Roman Kolpakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregory Kucherov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Computer Science, “A.I.Cuza” University, 6600, Iaşi, Romania

    Gabriel Ciobanu

  2. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700, Bucharest, Romania

    Gheorghe Păun

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kolpakov, R., Kucherov, G. (1999). On maximal repetitions in words. In: Ciobanu, G., Păun, G. (eds) Fundamentals of Computation Theory. FCT 1999. Lecture Notes in Computer Science, vol 1684. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48321-7_31

Download citation

  • DOI: https://doi.org/10.1007/3-540-48321-7_31

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66412-3

  • Online ISBN: 978-3-540-48321-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation