Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Combinatorial Algorithms

IWOCA 2010: Combinatorial Algorithms pp 10–19Cite as

On the Maximal Sum of Exponents of Runsin a String

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6460))

Abstract

A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p such that 2p ≤ |v|. The exponent of a run is defined as |v|/p and is ≥ 2. We show new bounds on the maximal sum of exponents of runs in a string of length n. Our upper bound of 4.1 n is better than the best previously known proven bound of 5.6 n by Crochemore & Ilie (2008). The lower bound of 2.035 n, obtained using a family of binary words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the maximal sum of exponents of runs in a string of length n is smaller than 2n.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berstel, J., Karhumaki, J.: Combinatorics on words: a tutorial. Bulletin of the EATCS 79, 178–228 (2003)

    MATH  Google Scholar 

  2. Crochemore, M., Ilie, L.: Analysis of maximal repetitions in strings. In: Kucera, L., Kucera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 465–476. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  3. Crochemore, M., Ilie, L.: Maximal repetitions in strings. J. Comput. Syst. Sci. 74(5), 796–807 (2008)

    Article  MATH  Google Scholar 

  4. Crochemore, M., Ilie, L., Rytter, W.: Repetitions in strings: Algorithms and combinatorics. Theor. Comput. Sci. 410(50), 5227–5235 (2009)

    Article  MATH  Google Scholar 

  5. Crochemore, M., Ilie, L., Tinta, L.: Towards a solution to the ”runs” conjecture. In: Ferragina, P., Landau, G.M. (eds.) CPM 2008. LNCS, vol. 5029, pp. 290–302. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  6. Crochemore, M., Iliopoulos, C.S., Kubica, M., Radoszewski, J., Rytter, W., Walen, T.: On the maximal number of cubic runs in a string. In: Dediu, A.H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 227–238. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Franek, F., Yang, Q.: An asymptotic lower bound for the maximal number of runs in a string. Int. J. Found. Comput. Sci. 19(1), 195–203 (2008)

    Article  MATH  Google Scholar 

  8. Giraud, M.: Not so many runs in strings. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 232–239. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  9. Gusfield, D., Stoye, J.: Simple and flexible detection of contiguous repeats using a suffix tree (preliminary version). In: Farach-Colton, M. (ed.) CPM 1998. LNCS, vol. 1448, pp. 140–152. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  10. Kolpakov, R.M., Kucherov, G.: Finding maximal repetitions in a word in linear time. In: Proceedings of the 40th Symposium on Foundations of Computer Science, pp. 596–604 (1999)

    Google Scholar 

  11. Kolpakov, R.M., Kucherov, G.: On maximal repetitions in words. J. of Discr. Alg. 1, 159–186 (1999)

    MATH  Google Scholar 

  12. Kolpakov, R.M., Kucherov, G.: On the sum of exponents of maximal repetitions in a word. Tech. Report 99-R-034, LORIA (1999)

    Google Scholar 

  13. Kusano, K., Matsubara, W., Ishino, A., Bannai, H., Shinohara, A.: New lower bounds for the maximum number of runs in a string. CoRR, abs/0804.1214 (2008)

    Google Scholar 

  14. Lothaire, M.: Combinatorics on Words. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  15. Puglisi, S.J., Simpson, J., Smyth, W.F.: How many runs can a string contain? Theor. Comput. Sci. 401(1-3), 165–171 (2008)

    Article  MATH  Google Scholar 

  16. Rytter, W.: The number of runs in a string: Improved analysis of the linear upper bound. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 184–195. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  17. Rytter, W.: The number of runs in a string. Inf. Comput. 205(9), 1459–1469 (2007)

    Article  MATH  Google Scholar 

  18. Simpson, J.: Modified Padovan words and the maximum number of runs in a word. Australasian J. of Comb. 46, 129–145 (2010)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. King’s College London, London, WC2R 2LS, UK

    Maxime Crochemore

  2. Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw, Poland

    Marcin Kubica, Jakub Radoszewski, Wojciech Rytter & Tomasz Waleń

  3. Université Paris-Est, France

    Maxime Crochemore

  4. Dept. of Math. and Informatics, Copernicus University, Toruń, Poland

    Wojciech Rytter

Authors
  1. Maxime Crochemore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcin Kubica
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jakub Radoszewski
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wojciech Rytter
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Tomasz Waleń
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of London, King’s College, The Strand, WC2R 2LS, London, UK

    Costas S. Iliopoulos

  2. Department of Computing and Software, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, ON, Canada

    William F. Smyth

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Crochemore, M., Kubica, M., Radoszewski, J., Rytter, W., Waleń, T. (2011). On the Maximal Sum of Exponents of Runsin a String. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2010. Lecture Notes in Computer Science, vol 6460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19222-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-19222-7_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-19221-0

  • Online ISBN: 978-3-642-19222-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation