Advertisement

A Generic Algorithm for Generating Closed Sets of a Binary Relation

  • Alain Gély
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3403)

Abstract

In this paper we propose a “divide and conquer” based generating algorithm for closed sets of a binary relation. We show that some existing algorithms are particular instances of our algorithm. This allows us to compare those algorithms and exhibit that the practical efficiency relies on the number of invalid closed sets generated. This number strongly depends on a choice function and the structure of the lattice. We exhibit a class of lattices for which no invalid closed sets are generated and thus reduce time complexity for such lattices. We made several tests which illustrate the impact of the choice function in practical efficiency.

Keywords

Generation algorithm Closure operator Galois or concept lattice 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, R., Imielinski, T., Swami, A.: Mining association rules between sets of items in large database. In: ACM SIGMOD conf. Management of data, pp. 265–290 (1993)Google Scholar
  2. 2.
    Bertet, K., Medina, R., Nourine, L., Raynaud, O.: Algorithmique combinatoire dans les bases de données massives. In: Actes du workshop ‘Usage des treillis de Galois pour l’intelligence artificielle’ AFIA 2003, Laval (2003)Google Scholar
  3. 3.
    Birkhoff, G.: Lattice Theory, third edition, vol. XXV. American Mathematical Colloquium Publications, Providence (1967)Google Scholar
  4. 4.
    Bordat, J.P.: Calcul pratique du treillis de galois d’une correspondance. Journal of Math. Sci. Hum. 96, 31–47 (1986)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Ganter, B.: Two basic algorithms in concept analysis. Technical report, Technische Hoschule Darmstadt (1984)Google Scholar
  6. 6.
    Ganter, B., Reuter, K.: Finding all closed sets: a general approach. Order 8 (1991)Google Scholar
  7. 7.
    Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  8. 8.
    Habib, M., Medina, R., Nourine, L., Steiner, G.: Efficient algorithms on distributive lattices. Discrete Applied Mathematics 110, 169–187 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kuznetsov, S., Obiedkov, S.: Comparing performance of algorithms for generating concept lattices. Journal of Experimental and Theoretical Artificial Intelligence (JETAI) 2/3(14), 189–216 (2002)CrossRefGoogle Scholar
  10. 10.
    Lindig, C.: Fast concept analysis. In: Stumme, G. (ed.) Working with Conceptual Structures, Contributions to ICCS 2000 (2000)Google Scholar
  11. 11.
    Markowsky, G.: Primes, irreducibles and extremal lattices. Order 9, 265–290 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nourine, L.: Une structuration algorithmique de la théorie des treillis. Habilitation à diriger des recherches, Université Montpellier II (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Alain Gély
    • 1
  1. 1.LIMOSUniversité Blaise PascalAubière CedexFrance

Personalised recommendations