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Computing on-line the lattice of maximal antichains of posets. (English) Zbl 0814.06004

Summary: We consider the on-line computation of the lattice of maximal antichains of a finite poset \(\widetilde{P}\). This on-line computation satisfies what we call the “linear extension hypothesis”: the new incoming vertex is always maximal in the current subposet of \(\widetilde{P}\). 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 \({\mathcal O}((| P| + \omega^ 2(P)) \omega(P)| MA(P)|)\), where \(| P|\) is the number of elements of the poset \(\widetilde{P}\), \(| MA(P)|\) is the number of maximal antichains of \(\widetilde{P}\) and \(\omega(P)\) is the width of \(\widetilde{P}\). This is more efficient than the best off-line algorithms known so far.

MSC:

06A06 Partial orders, general
68Q25 Analysis of algorithms and problem complexity
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[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.
[2] Birkhoff, G. (1937) Rings of sets,Duke Math. J-3, 311–316. · Zbl 0016.38702
[3] Behrendt, G. (1988) Maximal antichains in partially ordered sets,ARS Combinatoria 25C, 149–157. · Zbl 0657.06003
[4] Bonnet, R. and Pouzet, M. (1969) Extensions et stratifications d’ensembles dispersés, C.R.A.S. Paris, t. 268, Série A, 1512–1515. · Zbl 0188.04203
[5] Bordat, J. P. (1986) Calcul pratique du treillis de Galois d’une correspondance,Math. Sci. Hum. 96, 31–47. · Zbl 0626.06007
[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.
[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.
[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.
[9] Fidge, C. (1988) Timestamps in message passing systems that preserve the partial ordering, in:Proc. 11th Australian Computer Science Conference, pp. 55–66.
[10] Ganter, B. and Reuter, K. (1991) Finding all Closed sets: A general approach,Order 8, 283–290. · Zbl 0754.06003
[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.
[12] Irani, S. (1990) Coloring inductive graphs on-line,IEEE 31nd Symposium on Foundations of Computer Science pp. 470–479.
[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. · Zbl 0833.68055
[14] Lamport, L. (1978) Time, clocks and the ordering of events in a distributed system,Communications of the ACM 21(7), 558–565. · Zbl 0378.68027
[15] MacNeille, H. M. (1937) Partially ordered sets,Transactions of the American Mathematical Society 42, 416–460. · Zbl 0017.33904
[16] Markowsky, G. (1975) The factorization and representation of lattices,Transactions of the American Mathematical Society 203, 185–200. · Zbl 0302.06011
[17] Markowsky, G. (1992) Primes, irreducibles and extremal lattices,Order 9, 265–290. · Zbl 0778.06007
[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. · Zbl 0709.68611
[19] Morvan, M. and Nourine, L. (1992) Generating minimal interval extensions, R.R. No. 92-015, LIRMM Montpellier.
[20] Reuter, K. (1991) The jump number and the lattice of maximal antichains,Discrete Mathematics 88, 289–307. · Zbl 0731.06003
[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.
[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. · Zbl 0491.06008
[23] Wille, R. (1985) Finite distributive lattices as concept lattices,Atti. Inc. Logica Mathematica (Siena)2, 635–648. · Zbl 0577.06012
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