Skip to main content
SpringerLink
Log in

One-Variable Word Equations in Linear Time

  • Conference paper
Book cover Automata, Languages, and Programming (ICALP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7966))

Included in the following conference series:

Abstract

In this paper we consider word equations with one variable (and arbitrary many appearances of it). A recent technique of recompression, which is applicable to general word equations, is shown to be suitable also in this case. While in general case it is non-deterministic, it determinises in case of one variable and the obtained running time is \(\mathcal{O}(n)\) (in RAM model).

The full version of this paper is available at http://arxiv.org/abs/1302.3481

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 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berkman, O., Vishkin, U.: Recursive star-tree parallel data structure. SIAM J. Comput. 22(2), 221–242 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dąbrowski, R., Plandowski, W.: On word equations in one variable. Algorithmica 60(4), 819–828 (2011)

    Article  MathSciNet  Google Scholar 

  3. Jeż, A.: Faster fully compressed pattern matching by recompression. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 533–544. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  4. Jeż, A.: Approximation of grammar-based compression via recompression. In: Fischer, J., Sanders, P. (eds.) CPM 2013. LNCS, vol. 7922, pp. 165–176. Springer, Heidelberg (2013)

    Google Scholar 

  5. Jeż, A.: The complexity of compressed membership problems for finite automata. Theory of Computing Systems, 1–34 (2013), http://dx.doi.org/10.1007/s00224-013-9443-6

  6. Jeż, A.: Recompression: a simple and powerful technique for word equations. In: Portier, N., Wilke, T. (eds.) STACS. LIPIcs, vol. 20, pp. 233–244. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2013), http://drops.dagstuhl.de/opus/volltexte/2013/3937

    Google Scholar 

  7. Kärkkäinen, J., Sanders, P., Burkhardt, S.: Linear work suffix array construction. J. ACM 53(6), 918–936 (2006)

    Article  MathSciNet  Google Scholar 

  8. Kasai, T., Lee, G., Arimura, H., Arikawa, S., Park, K.: Linear-time longest-common-prefix computation in suffix arrays and its applications. In: Amir, A., Landau, G.M. (eds.) CPM 2001. LNCS, vol. 2089, pp. 181–192. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  9. Laine, M., Plandowski, W.: Word equations with one unknown. Int. J. Found. Comput. Sci. 22(2), 345–375 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lohrey, M., Mathissen, C.: Compressed membership in automata with compressed labels. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 275–288. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  11. Makanin, G.S.: The problem of solvability of equations in a free semigroup. Matematicheskii Sbornik 2(103), 147–236 (1977) (in Russian)

    Google Scholar 

  12. Mehlhorn, K., Sundar, R., Uhrig, C.: Maintaining dynamic sequences under equality tests in polylogarithmic time. Algorithmica 17(2), 183–198 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Obono, S.E., Goralcik, P., Maksimenko, M.N.: Efficient solving of the word equations in one variable. In: Privara, I., Ružička, P., Rovan, B. (eds.) MFCS 1994. LNCS, vol. 841, pp. 336–341. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  14. Plandowski, W.: Satisfiability of word equations with constants is in NEXPTIME. In: STOC, pp. 721–725 (1999)

    Google Scholar 

  15. Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. J. ACM 51(3), 483–496 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Plandowski, W., Rytter, W.: Application of Lempel-Ziv encodings to the solution of word equations. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 731–742. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  17. Sakamoto, H.: A fully linear-time approximation algorithm for grammar-based compression. J. Discrete Algorithms 3(2-4), 416–430 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute für Informatik, Campus E1 4, DE-66123, Saarbrücken, Germany

    Artur Jeż

  2. Institute of Computer Science, University of Wrocław, ul. Joliot-Curie 15, PL-50383, Wrocław, Poland

    Artur Jeż

Authors
  1. Artur Jeż
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

  2. Faculty of Computing, University of Latvia, Raina bulv. 19, 1586, Riga, Latvia

    Rūsiņš Freivalds

  3. Department of Computer Science, Wolfson Building, Parks Road, University of Oxford, OX1 3QD, Oxford, UK

    Marta Kwiatkowska

  4. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    David Peleg

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jeż, A. (2013). One-Variable Word Equations in Linear Time. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39212-2_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39212-2_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39211-5

  • Online ISBN: 978-3-642-39212-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation