Advertisement

On the Complexity of Hopcroft’s State Minimization Algorithm

  • Jean Berstel
  • Olivier Carton
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3317)

Abstract

Hopcroft’s algorithm for minimizing a deterministic automaton has complexity O(n log n). We show that this complexity bound is tight. More precisely, we provide a family of automata of size n = 2 k on which the algorithm runs in time k2 k . These automata have a very simple structure and are built over a one-letter alphabet. Their sets of final states are defined by de Bruijn words.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hopcroft, J.E., Ullman, J.D.: Formal Languages and their Relation to Automata. Addison-Wesley, Reading (1969)zbMATHGoogle Scholar
  2. 2.
    Hopcroft, J.E.: An n log n algorithm for minimizing states in a finite automaton. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 189–196. Academic Press, London (1971)Google Scholar
  3. 3.
    Krivol, S.L.: Algorithms for minimization of finite acyclic automata and pattern matching in terms. Cybernetics 27, 324–331 (1991); Translated from Kibernetika 3, 11–16 (May-June 1991)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Revuz, D.: Minimisation of acyclic deterministic automata in linear time. Theoret. Comput. Sci. 92, 181–189 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Daciuk, J.: Comparison of construction algorithms for minimal, acyclic, deterministic finite-state automata from sets of strings. In: Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2002. LNCS, vol. 2608, pp. 255–261. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Cardon, A., Crochemore, M.: Partitioning a graph in O(|A| log2 |V |). Theoret. Comput. Sci. 19, 85–98 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Paige, R., Tarjan, R.E., Bonic, R.: A linear time solution for the single function coarsest partition problem. Theoret. Comput. Sci. 40, 67–84 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 18, 973–989 (1987)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gai, A.T.: Algorithmes de partionnement : minimisation d’automates et applications aux graphes. Mémoire de DEA, Université Montpellier II (2003) Google Scholar
  10. 10.
    Gries, D.: Describing an algorithm by Hopcroft. Acta Inform. 2, 97–109 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Aho, A., Hopcroft, J., Ullman, J.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  12. 12.
    Beauquier, D., Berstel, J., Chrétienne, P.: Éléments d’algorithmique. Masson (1992)Google Scholar
  13. 13.
    Blum, N.: A O(n log n) implementation of the standard method for minimizing n-state finite automata. Inform. Proc. Letters 57, 65–69 (1996)zbMATHCrossRefGoogle Scholar
  14. 14.
    Knuutila, T.: Re-describing an algorithm by Hopcroft. Theoret. Comput. Sci. 250, 333–363 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tutte, W.T.: Graph Theory. Encyclopedia of Mathematics and its Applications, vol. 21. Addison-Wesley, Reading (1984)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jean Berstel
    • 1
  • Olivier Carton
    • 2
  1. 1.Institut Gaspard MongeUniversité de Marne-la-Vallée 
  2. 2.LIAFAUniversité Paris 7 

Personalised recommendations