A Fast Algorithm Finding the Shortest Reset Words

  • Andrzej Kisielewicz
  • Jakub Kowalski
  • Marek Szykuła
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7936)


In this paper we present a new fast algorithm for finding minimal reset words for finite synchronizing automata, which is a problem appearing in many practical applications. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional BFS and radix (Patricia) tries to store and compare subsets. Also a number of heuristics are applied. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with n ≤ 300 states and 2 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word \(\approx 2.5\sqrt{n-5}\).


Synchronizing automaton synchronizing word Černý conjecture 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ananichev, D., Gusev, V., Volkov, M.: Slowly synchronizing automata and digraphs. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 55–65. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Ananichev, D., Volkov, M.: Synchronizing monotonic automata. In: Ésik, Z., Fülöp, Z. (eds.) DLT 2003. LNCS, vol. 2710, pp. 111–121. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Benenson, Y., Adar, R., Paz-Elizur, T., Livneh, Z., Shapiro, E.: DNA molecule provides a computing machine with both data and fuel. Proceedings of the National Academy of Sciences 100(5), 2191–2196 (2003)CrossRefGoogle Scholar
  4. 4.
    Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Časopis Slovenskej Akadémie Vied 14(3), 208–216 (1964) (in Slovak)zbMATHGoogle Scholar
  5. 5.
    Chmiel, K., Roman, A.: COMPAS - A computing package for synchronization. In: Domaratzki, M., Salomaa, K. (eds.) CIAA 2010. LNCS, vol. 6482, pp. 79–86. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Devroye, L.: A note on the average depth of tries. Computing 28, 367–371 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eppstein, D.: Reset sequences for monotonic automata. SIAM Journal on Computing 19, 500–510 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gerbush, M., Heeringa, B.: Approximating minimum reset sequences. In: Domaratzki, M., Salomaa, K. (eds.) CIAA 2010. LNCS, vol. 6482, pp. 154–162. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Higgins, P.: The range order of a product of i-transformations from a finite full transformation semigroup. Semigroup Forum 37, 31–36 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jürgensen, H.: Synchronization. Information and Computation 206(9-10), 1033–1044 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kari, J.: Synchronization and stability of finite automata. Journal of Universal Computer Science 8(2), 270–277 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kudłacik, R., Roman, A., Wagner, H.: Effective synchronizing algorithms. Expert Systems with Applications 39(14), 11746–11757 (2012)CrossRefGoogle Scholar
  13. 13.
    Morrison, D.R.: PATRICIA – practical algorithm to retrieve information coded in alphanumeric. Journal of the ACM 15, 514–534 (1968)CrossRefGoogle Scholar
  14. 14.
    Olschewski, J., Ummels, M.: The complexity of finding reset words in finite automata. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 568–579. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Panneton, F., L’Ecuyer, P., Matsumoto, M.: Improved long-period generators based on linear recurrences modulo 2. ACM Transactions on Mathematical Software 32(1), 1–16 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Roman, A.: New algorithms for finding short reset sequences in synchronizing automata. In: International Enformatika Conference (Prague), pp. 13–17 (2005)Google Scholar
  17. 17.
    Roman, A.: Genetic algorithm for synchronization. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 684–695. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Sandberg, S.: Homing and synchronizing sequences. In: Broy, M., Jonsson, B., Katoen, J.-P., Leucker, M., Pretschner, A. (eds.) Model-Based Testing of Reactive Systems. LNCS, vol. 3472, pp. 5–33. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Skvortsov, E., Tipikin, E.: Experimental study of the shortest reset word of random automata. In: Bouchou-Markhoff, B., Caron, P., Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2011. LNCS, vol. 6807, pp. 290–298. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Skvortsov, E., Zaks, Y.: Synchronizing random automata. Discrete Mathematics and Theoretical Computer Science 12(4), 95–108 (2010)MathSciNetGoogle Scholar
  21. 21.
    Trahtman, A.N.: A package TESTAS for checking some kinds of testability. In: Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2002. LNCS, vol. 2608, pp. 228–232. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Trahtman, A.N.: An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 789–800. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Trahtman, A.N.: Modifying the upper bound on the length of minimal synchronizing word. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 173–180. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  24. 24.
    Volkov, M.V.: Synchronizing automata and the Černý conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Andrzej Kisielewicz
    • 1
    • 2
  • Jakub Kowalski
    • 1
  • Marek Szykuła
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of WrocławPoland
  2. 2.Institute of Mathematics and Computer ScienceUniversity of OpolePoland

Personalised recommendations