Decremental maintenance of reachability in hypergraphs and minimum models of horn formulae
In this paper we present a decremental algorithm for maintaining minimum rank hyperpaths in a directed hypergraph from a source vertex s to all other vertices, under the assumption of unit hyperedge weights. Given a hypergraph H with n vertices and m hyperedges, the total time needed to perform a sequence of m hyperedge deletions is O(n · Size(H)), where Size(H) is the sum of the sizes of the hyperedges of H; the total space needed is O(n + Size(H)). In the case of integer hyperedge weights in [1, C] our solution requires O(C · n · Size(H)) total time and O(C + n + Size(H)) space.
Using the algorithm presented in this paper, we also show how to maintain the satisfiability and the minimum model of a Horn formula F with n propositional symbols in total time O(n·Length(F)) over any sequence of clause deletions.
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- 1.Alimonti, P., Feuerstein, E., Nanni, U.: Linear time algorithms for liveness and boundedness in conflict-free petri nets. 1st Latin American Symposium on Theoretical Informatics, Säo Paulo, Brazil, April 1992; Lect. Notes in Comp. Sci., 583, 1–14, Springer-Verlag (April 1992).Google Scholar
- 2.Ausiello, G., D'Atri, A., Sacca, D.: Minimal representation of directed hypergraphs. SIAM Journal on Computing 2 (1986) 418–431.Google Scholar
- 3.Ausiello, G., Italiano, G. F.: On-line algorithms for polinomially solvable satisfiability problems. Journal of Logic Programming 10 (1991) 69–90.Google Scholar
- 4.Ausiello, G., Italiano, G. F., Nanni, U.: Dynamic maintenance of directed hypergraphs. Theoretical Computer Science 72 (1990) 97–117.Google Scholar
- 5.Ausiello, G., Italiano, G. F., Nanni, U.: Optimal traversal of directed hypergraphs. ICSI Technical Report TR-92-073 (September 1992).Google Scholar
- 6.Ausiello, G., D'Atri, A., Saccà, D.: Graph algorithms for functional dependency manipulation. J. ACM, 30, 752–766 (1983).Google Scholar
- 7.Ausiello, G., Giaccio, R.: On-line algorithms for satisfiability formulae with uncertainty. Theoretical Computer Science 171 (1997) 3–24.Google Scholar
- 8.Cambini, R., Gallo, G., Scutellà, M. G.: Flows on hypergraphs. To appear in Mathematical Programming B.Google Scholar
- 9.Dowling, W. F., Gallier, J. H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. Journal of Logic Programming, 3 (1984) 267–284.Google Scholar
- 10.Franciosa, P. G., Frigioni, D., Giaccio, R.: Semi-dynamic shortest paths and breadth first search in digraphs. Symposium on Theoretical Aspects of Computer Science 1997, LNCS vol. 1200, Springer-Verlag (February 1997) 33–46.Google Scholar
- 11.Gallo, G., Longo, G., Nguyen, S., Pallottino, S.: Directed hypergraphs and applications. Discrete Applied Mathematics 42 (1993) 177–201.Google Scholar
- 12.Italiano, G. F., Nanni, U.: On line maintenance of minimal directed hypergraphs. Proc. 3rd Italian Conference on Theoretical Computer Science, Mantova, Italy, 2–4 November 1989, World Scientific Co., 335–349.Google Scholar
- 13.Knuth, D. E.: A generalization of Dijkstra's algorithm, Information Processing Letters 6 (1) pp. 1–5 (1977).Google Scholar
- 14.Ramalingam, G., Reps, T.: An incremental algorithm for a generalized shortest-path problem. Journal of Algorithms 21 (1996) 267–305.Google Scholar