Computing All ℓ-Cover Automata Fast

  • Artur Jeż
  • Andreas Maletti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6807)


Given a language L and a number ℓ, an ℓ-cover automaton for L is a DFA M such that its language coincides with L on all words of length at most ℓ. It is known that an equivalent minimal ℓ-cover automaton can be constructed in time \(\mathcal{O}(n \log n)\), where n is the number of states of M. This is achieved by a clever and sophisticated variant of Hopcroft’s algorithm, which computes the ℓ-similarity inside the main algorithm. This contribution presents an alternative simple algorithm with running time \(\mathcal{O}(n \log n)\), in which the computation is split into three phases. First, a compact representation of the gap table is created. Second, this representation is enriched with information about the length of a shortest word leading to the states. These two steps are independent of the parameter ℓ. Third, the ℓ-similarity is extracted by simple comparisons against ℓ. In particular, this approach allows the calculation of all the sizes of minimal ℓ-cover automata (for all valid ℓ) in the same time bound.


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  1. 1.
    Badr, A., Geffert, V., Shipman, I.: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theoret. Inform. Appl. 43(1), 69–94 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Câmpeanu, C., Paun, A., Yu, S.: An efficient algorithm for constructing minimal cover automata for finite languages. Int. J. Found. Comput. Sci. 13(1), 83–97 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Câmpeanu, C., Santean, N., Yu, S.: Minimal cover-automata for finite languages. Theor. Comput. Sci. 267(1-2), 3–16 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Champarnaud, J.-M., Guingne, F., Hansel, G.: Similarity relations and cover automata. RAIRO Theoret. Inform. Appl. 39(1), 115–123 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gawrychowski, P., Jeż, A.: Hyper-minimisation made efficient. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 356–368. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Gawrychowski, P., Jeż, A., Maletti, A.: On minimising automata with errors. Corr. abs/1102.5682 (2011)Google Scholar
  7. 7.
    Gries, D.: Describing an algorithm by Hopcroft. Acta Inf. 2(2), 97–109 (1973)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hartigan, J.A.: Representation of similarity matrices by trees. J. Amer. Statist. Assoc. 62(320), 1140–1158 (1967)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hopcroft, J.E.: An \(n\,\textrm{log}\, n\) algorithm for minimizing states in a finite automaton. In: Kohavi, Z. (ed.) Theory of Machines and Computations, pp. 189–196. Academic Press, London (1971)CrossRefGoogle Scholar
  10. 10.
    Jardine, C.J., Jardine, N., Sibson, R.: The structure and construction of taxonomic hierarchies. Math. Biosci. 1(2), 173–179 (1967)CrossRefMATHGoogle Scholar
  11. 11.
    Johnson, S.C.: Hierarchical clustering schemes. Psychometrika 32(3), 241–254 (1967)CrossRefGoogle Scholar
  12. 12.
    Körner, H.: A time and space efficient algorithm for minimizing cover automata for finite languages. Int. J. Found. Comput. Sci. 14(6), 1071–1086 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Schewe, S.: Beyond hyper-minimisation — Minimising DBAs and DPAs is NP-complete. In: Proc. Ann. Conf. Foundations of Software Technology and Theoretical Computer Science, LIPIcs, vol. 8, pp. 400–411. Schloss Dagstuhl (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Artur Jeż
    • 1
  • Andreas Maletti
    • 2
  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland
  2. 2.Institute for Natural Language ProcessingUniversität StuttgartStuttgartGermany

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