Abstract
Several efficient transitive closure algorithms operate on the strongly connected components of a digraph, some of them using Tarjan's algorithm [17]. Exploiting facts from graph theory and the special properties of Tarjan's algorithm we develop a new, improved algorithm. The transitive reduction of a digraph defined in [1] may be obtained as a byproduct.
Zusammenfassung
Verschiedene effiziente Transitive-Hülle-Algorithmen arbeiten auf den stark zusammenhängenden Komponenten eines gerichteten Graphen; einige davon verwenden den Algorithmus von Tarjan [17]. Wir nützen Sachverhalte aus der Graphentheorie und die speziellen Eigenschaften von Tarjans Algorithmus aus, um einen neuen, verbesserten Algorithmus zu entwickeln. Die transitive Reduktion eines gerichteten Graphen, wie sie in [1] definiert wird, läßt sich als Nebenprodukt bestimmen.
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References
Aho, A. V., Garey, M. R., Ullmann, J. D.: The transitive reduction of a directed graph. SIAM J. Comput.1, 131–137 (1972).
Arlazarov, V. L., Dinic, E. A., Kronrod, M. A., Faradzev, I. A.: On economical construction of the transitive closure of an oriented graph. Soviet Math. Dokl.11, 1209–1210 (1970).
Bayer, R.: Aggregates: A software design method and its application to a family of transitive closure algorithms. Technische Universität München, 1974.
Bloniarz, P. A., Fischer, M. J., Meyer, A. R.: A note on the average time to compute transitive closures. In: Automata, Languages and Programming (Michaelson, Milner, eds.). Edinburgh: Edinburgh University Press 1976.
Coffy, J.: On computing the time complexity of transitive closure algorithms. Inf. Proc. Letters2, 39–42 (1973).
Dijkstra, E. W.: A discipline of programming. Englewood Cliffs, N. J.: Prentice-Hall 1976.
Dzikiewicz, J.: An algorithm for finding the transitive closure of a digraph. Computing15, 75–79 (1975).
Ebert, J.: A sensitive transitive closure algorithm. Inf. Proc. Letters12, 255–258 (1981).
Eve, J., Kurki-Suonio, R.: On computing the transitive closure of a relation. Acta Informatica8, 303–314 (1977).
Furman, M. E.: Applications of a method of fast multiplication of matrices in the problem of finding the transitive closure of a graph. Soviet Math. Dokl.11, 1252 (1970).
Harary, F., Norman, R. Z., Cartwright, D.: Structured models: An introduction to the theory of directed graphs. New York: Wiley 1965.
Munro, I.: Efficient determination of the transitive closure of a directed graph. Inf. Proc. Letters1, 56–58 (1971).
O'Neil, P. E., O'Neil, E. J.: A fast expected time algorithm for Boolean matrix multiplication and transitive closure. In: Courant Computer Science Symp. 9: Combinatorial algorithms (Rustin, ed.). New York: Algorithmics Press 1973.
Purdom, P.: A transitive closure algorithm. BIT10, 76–94 (1970).
Schnorr, C. P.: An algorithm for transitive closure with linear expected time. SIAM J. Comput.7, 127–133 (1978).
Syslo, M. M., Dzikiewicz, J.: Computational experiences with some transitive closure algorithms. Computing15, 33–39 (1975).
Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput.1, 146–160 (1972).
Warshall, S.: A theorem on Boolean matrices. J. ACM1962, 11–12.
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Schmitz, L. An improved transitive closure algorithm. Computing 30, 359–371 (1983). https://doi.org/10.1007/BF02242140
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DOI: https://doi.org/10.1007/BF02242140