Order

, Volume 11, Issue 3, pp 197–210 | Cite as

Computing on-line the lattice of maximal antichains of posets

  • Guy-Vincent Jourdan
  • Jean-Xavier Rampon
  • Claude Jard
Article

Abstract

We consider the on-line computation of the lattice of maximal antichains of a finite poset\(\tilde P\). This on-line computation satisfies what we call the “linear extension hypothesis”: the new incoming vertex is always maximal in the current subposet of\(\tilde P\). In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of\(\mathcal{O}((\left| P \right| + \omega ^2 (P))\omega (P)\left| {MA(P)} \right|),\), where |P| is the number of elements of the poset,\(\tilde P\), |MA(P)| is the number of maximal antichains of\(\tilde P\) and ω(P) is the width of\(\tilde P\). This is more efficient than the best off-line algorithms known so far.

Mathematics Subject Classifications (1991)

06A10 68C25 

Key words

On-line algorithm lattice of maximal antichains 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Babaoglu, O. and Raynal, M. (1993)Sequence-Based Global Predicates for Distributed Computations: Definitions and Detection Algorithms, IRISA research report No. 729, May 1993.Google Scholar
  2. 2.
    Birkhoff, G. (1937) Rings of sets,Duke Math. J-3, 311–316.CrossRefGoogle Scholar
  3. 3.
    Behrendt, G. (1988) Maximal antichains in partially ordered sets,ARS Combinatoria 25C, 149–157.Google Scholar
  4. 4.
    Bonnet, R. and Pouzet, M. (1969) Extensions et stratifications d'ensembles dispersés, C.R.A.S. Paris, t. 268, Série A, 1512–1515.Google Scholar
  5. 5.
    Bordat, J. P. (1986) Calcul pratique du treillis de Galois d'une correspondance,Math. Sci. Hum. 96, 31–47.Google Scholar
  6. 6.
    Charron-Bost, B., Delporte-Gallet, C. and Fauconnier, H. (1992) Local and temporal predicates in distributed systems,Research Report No. 92-36, LITP, Paris 7.Google Scholar
  7. 7.
    Cooper, R. and Marzullo, K. (1991) Consistent detection of global predicates, in:Proc. ACM/ONR Workshop on Parallel and Distributed Debugging, pp. 163–173, Santa Cruz, California.Google Scholar
  8. 8.
    Diehl, C., Jard, C. and Rampon, J. X. (1993) Reachablity analysis on distributed executions, TAPSOFT'93: Theory and Practice of Software Development, in Lecture Notes in Computer Science No. 668, Springer-Verlag, pp. 629–643.Google Scholar
  9. 9.
    Fidge, C. (1988) Timestamps in message passing systems that preserve the partial ordering, in:Proc. 11th Australian Computer Science Conference, pp. 55–66.Google Scholar
  10. 10.
    Ganter, B. and Reuter, K. (1991) Finding all Closed sets: A general approach,Order 8, 283–290.CrossRefGoogle Scholar
  11. 11.
    Habib, M., Morvan, M., Pouzet, M. and Rampon, J. X. (1992) Incidence, structures, coding and lattice of maximal antichains, Research Report No. 92-079, LIRMM Montpellier.Google Scholar
  12. 12.
    Irani, S. (1990) Coloring inductive graphs on-line,IEEE 31nd Symposium on Foundations of Computer Science pp. 470–479.Google Scholar
  13. 13.
    Jard, C., Jourdan, G. V. and Rampon, J. X. (1993) Some “On-line” computations of the ideal lattice of posets,IRISA Research Report No. 773.Google Scholar
  14. 14.
    Lamport, L. (1978) Time, clocks and the ordering of events in a distributed system,Communications of the ACM 21(7), 558–565.CrossRefGoogle Scholar
  15. 15.
    MacNeille, H. M. (1937) Partially ordered sets,Transactions of the American Mathematical Society 42, 416–460.MathSciNetGoogle Scholar
  16. 16.
    Markowsky, G. (1975) The factorization and representation of lattices,Transactions of the American Mathematical Society 203, 185–200.Google Scholar
  17. 17.
    Markowsky, G. (1992) Primes, irreducibles and extremal lattices,Order 9, 265–290.CrossRefGoogle Scholar
  18. 18.
    Mattern, F. (1989) Virtual time and global states of distributed systems, in: Cosnard, Quinton, Raynal and Robert, (eds.),Proc. Int. Workshop on Parallel Distributed Algorithms, Bonas France, North-Holland.Google Scholar
  19. 19.
    Morvan, M. and Nourine, L. (1992) Generating minimal interval extensions, R.R. No. 92-015, LIRMM Montpellier.Google Scholar
  20. 20.
    Reuter, K. (1991) The jump number and the lattice of maximal antichains,Discrete Mathematics 88, 289–307.CrossRefGoogle Scholar
  21. 21.
    Westbrook, J. and Yan, D. C. K. (1993) Greedy Algorithms for the On-Line Steiner Tree and Generalized Steiner Problems, WADS'93:Algorithms and Data Structures, Lecture Notes in Computer Science No. 709, Springer-Verlag, pp. 622–633.Google Scholar
  22. 22.
    Wille, R. (1982) Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (ed.),Ordered Sets, Reidel, Dordrecht, pp. 445–470.Google Scholar
  23. 23.
    Wille, R. (1985) Finite distributive lattices as concept lattices,Atti. Inc. Logica Mathematica (Siena)2, 635–648.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Guy-Vincent Jourdan
    • 1
  • Jean-Xavier Rampon
    • 1
  • Claude Jard
    • 1
  1. 1.IRISARennesFrance

Personalised recommendations