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.
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Rakesh Agrawal, Alex Borgida, and H.V. Jagadish. Efficient management of transitive relationships in large data bases, including is-a hierarchies. ACM SIGMOD, 1989.
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.
G. Behrendt. Maximal antichains in partially ordered sets. Ars Combin., C(25):149–157, 1988.
G. Birkhoff. Lattice Theory, volume 25 of Coll. Publ. XXV. American Mathematical Society, Providence, 3rd edition, 1967.
J.P. Bordat. Calcul pratique du treillis de gallois d'une correspondance. In Math. Sci. Hum, 96, pages 31–47, 1986.
A. Bouchet. Codages et dimensions de relations binaires. Annals of Discrete Mathematics 23, Ordres: Description and Roles, (M. Pouzet, D. Richard eds), 1984.
Yves Caseau. Efficient handling of multiple inheritance hierarchies. In OOPSLA '93, pages 271–287, 1993.
B. Charron-Bost. Mesures de la Concurrence et du Parallélisme des Calculs Répartis. PhD thesis, Université Paris VII, Paris, France, Septembre 1989.
B. A. Davey and H. A. Priestley. Introduction to lattices and orders. Cambridge University Press, second edition, 1991.
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.
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.
G. Gambosi, J. Nesetril, and M. Talamo. Efficient representation of taxonomies. In TAP-SOFT, CAAP Conf. Pisa, pages 232–240, 1987.
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.
G. Gambosi, J. Nesetril, and M. Talamo. On locally presented posets. Theoretical Comp. Sci., 3(70):251–260, 1990.
R. Godin and H. Mili. Building and maintening analysis-level class hierarchies using galois lattices. In OOPSLA '93, pages 394–410, 1993.
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.
M. Habib, M. Morvan, M. Pouzet, and J.-X. Rampon. Extensions intervallaires minimales. C. R. Acad. Sci. Paris, I(313):893–898, 1991.
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.
M. Habib and L. Nourine. Tree structure for distributive lattices and its applications. Research report, LIRMM, Montpellier, France, Avril 1994.
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.
G. Markowsky. Some combinatorial aspects of lattice theory. In Houston Lattice Theory Conf., editor, Proc. Univ. of Houston, pages 36–68, 1973.
G. Markowsky. The factorization and representation of lattices. Trans. of Amer. Math. Soc., 203:185–200, 1975.
G. Markowsky. Primes, irreducibles and extremal lattices. Order, 9:265–290, 1992.
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.
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.
L. Nourine. Quelques propriétés algorithmiques des treillis. PhD thesis, Université Montpellier II, Montpellier, France, June 1993.
K. Reuter. The jump number and the lattice of maximal antichains. Discrete Mathematics, 1991.
J. Stahl and R. Wille. Preconcepts of contexts. in Proc. Universal Algebra (Sienna), 1984.
W.T. Trotter. Combinatorics and Partially Ordered Sets: Dimension Theory. Press, Baltimore. John Hopkins University, 1992.
R. Wille. Restructuring lattice theory. in Ordered sets, I. Rival, Eds. NATO ASI No 83, Reidel, Dordecht, Holland, 1982.
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Habib, M., Nourine, L. (1994). Bit-vector encoding for partially ordered sets. In: Bouchitté, V., Morvan, M. (eds) Orders, Algorithms, and Applications. ORDAL 1994. Lecture Notes in Computer Science, vol 831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0019423
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DOI: https://doi.org/10.1007/BFb0019423
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