Abstract
For a given collection \(\mathcal{G}\) of directed graphs we define the join-reachability graph of \(\mathcal{G}\), denoted by \(\mathcal{J}(\mathcal{G})\), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of \(\mathcal{G}\). Our goal is to compute an efficient representation of \(\mathcal{J}(\mathcal{G})\). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for \(\mathcal{G}\). In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of \(\mathcal{G}\). This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.
This research project has been funded by the John S. Latsis Public Benefit Foundation. The sole responsibility for the content of this paper lies with its authors.
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References
Agrawal, R., Borgida, A., Jagadish, H.V.: Efficient management of transitive relationships in large data and knowledge bases. In: SIGMOD 1989: Proceedings of the 1989 ACM SIGMOD International Conference on Management of Data, pp. 253–262 (1989)
Aho, A.V., Garey, M.R., Ullman, J.D.: The transitive reduction of a directed graph. SIAM J. Comput. 1(2), 131–137 (1972)
Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation methods for the web. In: WWW 2001: Proceedings of the 10th International Conference on World Wide Web, pp. 613–622 (2001)
Georgiadis, L.: Computing frequency dominators and related problems. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 704–715. Springer, Heidelberg (2008)
Georgiadis, L.: Testing 2-vertex connectivity and computing pairs of vertex-disjoint s-t paths in digraphs. In: Proc. 37th Int’l. Coll. on Automata, Languages, and Programming, pp. 738–749 (2010)
Georgiadis, L., Nikolopoulos, S.D., Palios, L.: Join-reachability problems in directed graphs. Technical Report arXiv:1012.4938v1 [cs.DS] (2010)
Georgiadis, L., Tarjan, R.E.: Dominator tree verification and vertex-disjoint paths. In: Proc. 16th ACM-SIAM Symp. on Discrete Algorithms, pp. 433–442 (2005)
Kameda, T.: On the vector representation of the reachability in planar directed graphs. Information Processing Letters 3(3), 75–77 (1975)
Katriel, I., Kutz, M., Skutella, M.: Reachability substitutes for planar digraphs. Technical Report MPI-I-2005-1-002, Max-Planck-Institut Für Informatik (2005)
Talamo, M., Vocca, P.: An efficient data structure for lattice operations. SIAM J. Comput. 28(5), 1783–1805 (1999)
Tamassia, R., Tollis, I.G.: Dynamic reachability in planar digraphs with one source and one sink. Theoretical Computer Science 119(2), 331–343 (1993)
Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM 51(6), 993–1024 (2004)
Wang, H., He, H., Yang, J., Yu, P.S., Yu, J.X.: Dual labeling: Answering graph reachability queries in constant time. In: ICDE 2006: Proceedings of the 22nd International Conference on Data Engineering, p. 75 (2006)
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Georgiadis, L., Nikolopoulos, S.D., Palios, L. (2011). Join-Reachability Problems in Directed Graphs. In: Kulikov, A., Vereshchagin, N. (eds) Computer Science – Theory and Applications. CSR 2011. Lecture Notes in Computer Science, vol 6651. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20712-9_15
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DOI: https://doi.org/10.1007/978-3-642-20712-9_15
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