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Bit-vector encoding for partially ordered sets

  • Michel Habib
  • Lhouari Nourine
Invited Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 831)

Abstract

Given a lattice L, we propose a tree representation of L. We show that this tree contains a bit-vector encoding of L and then how to compute from this tree the lattice operations (meet and join). Algorithms which provide bit-vectors encodings for partial orders have been recently proposed in the literature. Given a partial order P we recall that computing an optimal bit-vector encoding of P is NP-Complete. From a theoretical lattice point of view we propose bit-vector encodings and study their optimality. We end by suggesting a data structure (lazy MacNeille completion) which can have many applications.

Keywords

Lattice theory Galois lattice MacNeille Completion Distributive Lattice and Lattice of Ideals Bit Vector Encoding Bounded Dimension 

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References

  1. 1.
    Rakesh Agrawal, Alex Borgida, and H.V. Jagadish. Efficient management of transitive relationships in large data bases, including is-a hierarchies. ACM SIGMOD, 1989.Google Scholar
  2. 2.
    Hassan Aït-Kaci, Robert Boyer, Patrick Lincoln, and Roger Nasr. Efficient implementation of lattice operations. ACM Transactions on Programming Langages and Systems, 11(1):115–146, January 1989.Google Scholar
  3. 3.
    G. Behrendt. Maximal antichains in partially ordered sets. Ars Combin., C(25):149–157, 1988.Google Scholar
  4. 4.
    G. Birkhoff. Lattice Theory, volume 25 of Coll. Publ. XXV. American Mathematical Society, Providence, 3rd edition, 1967.Google Scholar
  5. 5.
    J.P. Bordat. Calcul pratique du treillis de gallois d'une correspondance. In Math. Sci. Hum, 96, pages 31–47, 1986.Google Scholar
  6. 6.
    A. Bouchet. Codages et dimensions de relations binaires. Annals of Discrete Mathematics 23, Ordres: Description and Roles, (M. Pouzet, D. Richard eds), 1984.Google Scholar
  7. 7.
    Yves Caseau. Efficient handling of multiple inheritance hierarchies. In OOPSLA '93, pages 271–287, 1993.Google Scholar
  8. 8.
    B. Charron-Bost. Mesures de la Concurrence et du Parallélisme des Calculs Répartis. PhD thesis, Université Paris VII, Paris, France, Septembre 1989.Google Scholar
  9. 9.
    B. A. Davey and H. A. Priestley. Introduction to lattices and orders. Cambridge University Press, second edition, 1991.Google Scholar
  10. 10.
    G. Ellis. Efficient retrieval from hierarchies of objects using lattice operations. In Conceptual Graphs for knowledge representation, (Proc. International conference on Conceptual Structures, Quebec City, Canada, August 4–7, 1993), G. W. Mineau, B. Moulin and J. Sowa, Eds, Lecture Notes in Artificial Intelligence 699, Springer, Berlin, 1993.Google Scholar
  11. 11.
    G. Ellis and F. Lehmann. Exploiting the induced order on type-labeled graphs for fast knowledge retrieval. In Proc. of the 2nd International conference on Conceptual Structures, August 16–20, 1994), College Park, Maryland, Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, 1994.Google Scholar
  12. 12.
    G. Gambosi, J. Nesetril, and M. Talamo. Efficient representation of taxonomies. In TAP-SOFT, CAAP Conf. Pisa, pages 232–240, 1987.Google Scholar
  13. 13.
    G. Gambosi, J. Nesetril, and M. Talamo. Posets, boolean representations and quick path searching. In Proc. of the 14th International colloque on Automata, Languages and Programming, Lecture Notes in Computer Science 267, Springer-Verlag, Berlin, 1987.Google Scholar
  14. 14.
    G. Gambosi, J. Nesetril, and M. Talamo. On locally presented posets. Theoretical Comp. Sci., 3(70):251–260, 1990.Google Scholar
  15. 15.
    R. Godin and H. Mili. Building and maintening analysis-level class hierarchies using galois lattices. In OOPSLA '93, pages 394–410, 1993.Google Scholar
  16. 16.
    J.R. Griggs, J. Stahl, and W.T. Trotter. A sperner theorem on unrelated chains of subsets. J. Comb. theory. (A), pages 124–127, 1984.Google Scholar
  17. 17.
    M. Habib, M. Morvan, M. Pouzet, and J.-X. Rampon. Extensions intervallaires minimales. C. R. Acad. Sci. Paris, I(313):893–898, 1991.Google Scholar
  18. 18.
    M. Habib and L. Nourine. A linear time algorithm to recognize distributive lattices. Research Report 92-012, submitted to Order, LIRMM, Montpellier, France, March 1993.Google Scholar
  19. 19.
    M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Research report, LIRMM, Montpellier, France, Avril 1994.Google Scholar
  20. 20.
    C. Jard, G.-V. Jourdan, and J.-X. Rampon. Computing on-line the lattice of maximal antichains of posets. Thechnical report, IRISA, Rennes, France, February 1994.Google Scholar
  21. 21.
    G. Markowsky. Some combinatorial aspects of lattice theory. In Houston Lattice Theory Conf., editor, Proc. Univ. of Houston, pages 36–68, 1973.Google Scholar
  22. 22.
    G. Markowsky. The factorization and representation of lattices. Trans. of Amer. Math. Soc., 203:185–200, 1975.Google Scholar
  23. 23.
    G. Markowsky. Primes, irreducibles and extremal lattices. Order, 9:265–290, 1992.Google Scholar
  24. 24.
    F. Mattern. Virtual time and global states of distributed systems. In M. Cosnard and al., editors, Parallel and Distributed Algorithms, pages 215–226. Elsevier / North-Holland, 1989.Google Scholar
  25. 25.
    M. Morvan and L. Nourine. Sur la distributivité du treillis des antichaînes maximales d'un ensemble ordonné. C.R. Acad. Sci., t. 317-Série I:129–133, 1993.Google Scholar
  26. 26.
    L. Nourine. Quelques propriétés algorithmiques des treillis. PhD thesis, Université Montpellier II, Montpellier, France, June 1993.Google Scholar
  27. 27.
    K. Reuter. The jump number and the lattice of maximal antichains. Discrete Mathematics, 1991.Google Scholar
  28. 28.
    J. Stahl and R. Wille. Preconcepts of contexts. in Proc. Universal Algebra (Sienna), 1984.Google Scholar
  29. 29.
    W.T. Trotter. Combinatorics and Partially Ordered Sets: Dimension Theory. Press, Baltimore. John Hopkins University, 1992.Google Scholar
  30. 30.
    R. Wille. Restructuring lattice theory. in Ordered sets, I. Rival, Eds. NATO ASI No 83, Reidel, Dordecht, Holland, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Michel Habib
    • 1
  • Lhouari Nourine
    • 1
  1. 1.Département d'Informatique Fondamentale LIRMMUniversité Montpellier IIMontpellier Cedex 5France

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