CSR 2011: Computer Science – Theory and Applications pp 195-208 | Cite as
Join-Reachability Problems in Directed Graphs
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.
Keywords
Span Tree Directed Graph Unoriented Tree Query Time Reachability ProblemPreview
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References
- 1.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)Google Scholar
- 2.Aho, A.V., Garey, M.R., Ullman, J.D.: The transitive reduction of a directed graph. SIAM J. Comput. 1(2), 131–137 (1972)MathSciNetCrossRefMATHGoogle Scholar
- 3.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)Google Scholar
- 4.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)CrossRefGoogle Scholar
- 5.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)Google Scholar
- 6.Georgiadis, L., Nikolopoulos, S.D., Palios, L.: Join-reachability problems in directed graphs. Technical Report arXiv:1012.4938v1 [cs.DS] (2010)Google Scholar
- 7.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)Google Scholar
- 8.Kameda, T.: On the vector representation of the reachability in planar directed graphs. Information Processing Letters 3(3), 75–77 (1975)MathSciNetCrossRefMATHGoogle Scholar
- 9.Katriel, I., Kutz, M., Skutella, M.: Reachability substitutes for planar digraphs. Technical Report MPI-I-2005-1-002, Max-Planck-Institut Für Informatik (2005)Google Scholar
- 10.Talamo, M., Vocca, P.: An efficient data structure for lattice operations. SIAM J. Comput. 28(5), 1783–1805 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 11.Tamassia, R., Tollis, I.G.: Dynamic reachability in planar digraphs with one source and one sink. Theoretical Computer Science 119(2), 331–343 (1993)MathSciNetCrossRefMATHGoogle Scholar
- 12.Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM 51(6), 993–1024 (2004)MathSciNetCrossRefMATHGoogle Scholar
- 13.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)Google Scholar