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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 u and v, contains a path from u to v 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 problems. 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.

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Notes

  1. 1.

    We caution the reader not to confuse the terms “unoriented” and “undirected”.

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Acknowledgements

We would like to thank Li Zhang for several useful discussions. We also thank the anonymous reviewers for helpful comments and suggestions.

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Correspondence to Loukas Georgiadis.

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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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Georgiadis, L., Nikolopoulos, S.D. & Palios, L. Join-Reachability Problems in Directed Graphs. Theory Comput Syst 55, 347–379 (2014). https://doi.org/10.1007/s00224-013-9450-7

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Keywords

  • Algorithms
  • Data structures
  • Graph algorithms
  • Reachability
  • Combinatorial complexity