Border Algorithms for Computing Hasse Diagrams of Arbitrary Lattices

  • José L. Balcázar
  • Cristina Tîrnăucă
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6628)


The Border algorithm and the iPred algorithm find the Hasse diagrams of FCA lattices. We show that they can be generalized to arbitrary lattices. In the case of iPred, this requires the identification of a join-semilattice homomorphism into a distributive lattice.


Lattices Hasse diagrams border algorithms 


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  1. 1.
    Aggarwal, C.C., Yu, P.S.: A new approach to online generation of association rules. IEEE Transactions on Knowledge and Data Engineering 13(4), 527–540 (2001)CrossRefGoogle Scholar
  2. 2.
    Baixeries, J.: Lattice Characterization of Armstrong and Symmetric Dependencies. Ph.D. thesis, Universitat Politècnica de Catalunya (2007)Google Scholar
  3. 3.
    Baixeries, J.: A formal context for symmetric dependencies. In: Medina and Obiedkov [15], pp. 90–105Google Scholar
  4. 4.
    Baixeries, J., Balcázar, J.L.: Unified characterization of symmetric dependencies with lattices. In: Ganter, B., Kwuida, L. (eds.) Contributions to the 4th International Conference on Formal Concept Analysis (ICFCA). Verlag Allgemeine Wissensch (2006)Google Scholar
  5. 5.
    Baixeries, J., Szathmary, L., Valtchev, P., Godin, R.: Yet a faster algorithm for building the Hasse diagram of a concept lattice. In: Ferré, S., Rudolph, S. (eds.) ICFCA 2009. LNCS, vol. 5548, pp. 162–177. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Balcázar, J.L.: Redundancy, deduction schemes, and minimum-size bases for association rules. Logical Methods in Computer Science 6(2:3), 1–33 (2010)MathSciNetMATHGoogle Scholar
  7. 7.
    Davey, B., Priestley, H.: Introduction to Lattices and Orders, 2nd edn. Cambridge University Press, Cambridge (1991)MATHGoogle Scholar
  8. 8.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Godin, R., Missaoui, R., Alaoui, H.: Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence 11, 246–267 (1995)CrossRefGoogle Scholar
  10. 10.
    Guigues, J., Duquenne, V.: Familles minimales d’implications informatives résultant d’un tableau de données binaires. Mathématiques et Sciences Humaines 95, 5–18 (1986)Google Scholar
  11. 11.
    Kryszkiewicz, M.: Representative association rules. In: Wu, X., Ramamohanarao, K., Korb, K.B. (eds.) PAKDD 1998. LNCS (LNAI), vol. 1394, pp. 198–209. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Kuznetsov, S.O., Obiedkov, S.A.: Algorithms for the construction of concept lattices and their diagram graphs. In: Raedt, L.D., Siebes, A. (eds.) PKDD 2001. LNCS (LNAI), vol. 2168, pp. 289–300. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Luxenburger, M.: Implications partielles dans un contexte. Mathématiques et Sciences Humaines 29, 35–55 (1991)MathSciNetMATHGoogle Scholar
  14. 14.
    Martin, B., Eklund, P.W.: From concepts to concept lattice: A border algorithm for making covers explicit. In: Medina and Obiedkov [15], pp. 78–89Google Scholar
  15. 15.
    Medina, R., Obiedkov, S. (eds.): ICFCA 2008. LNCS (LNAI), vol. 4933. Springer, Heidelberg (2008)Google Scholar
  16. 16.
    Nourine, L., Raynaud, O.: A fast algorithm for building lattices. Information Processing Letters 71(5-6), 199–204 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Sagiv, Y., Delobel, C., Parker Jr., D.S., Fagin, R.: An equivalence between relational database dependencies and a fragment of propositional logic. Journal of the ACM 28(3), 435–453 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sagiv, Y., Delobel, C., Parker Jr., D.S., Fagin, R.: Correction to “An equivalence between relational database dependencies and a fragment of propositional logic”. Journal of the ACM 34(4), 1016–1018 (1987)CrossRefMATHGoogle Scholar
  19. 19.
    Savnik, I., Flach, P.A.: Discovery of multivalued dependencies from relations. Intelligent Data Analysis 4(3-4), 195–211 (2000)MATHGoogle Scholar
  20. 20.
    Valtchev, P., Missaoui, R., Lebrun, P.: A fast algorithm for building the Hasse diagram of a Galois lattice. In: Leroux, P. (ed.) Publications du LaCIM, pp. 293–306 (2000)Google Scholar
  21. 21.
    Zaki, M.J., Hsiao, C.J.: Efficient algorithms for mining closed itemsets and their lattice structure. IEEE Transactions on Knowledge and Data Engineering 17(4), 462–478 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • José L. Balcázar
    • 1
  • Cristina Tîrnăucă
    • 1
  1. 1.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain

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