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.


Minimisation Algorithm Compact Representation Short Word Outgoing Transition Incoming Transition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Câmpeanu, C., Santean, N., Yu, S.: Minimal cover-automata for finite languages. Theor. Comput. Sci. 267(1-2), 3–16 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Champarnaud, J.-M., Guingne, F., Hansel, G.: Similarity relations and cover automata. RAIRO Theoret. Inform. Appl. 39(1), 115–123 (2005)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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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