Advertisement

Algorithmica

pp 1–18 | Cite as

Reachability Oracles for Directed Transmission Graphs

  • Haim Kaplan
  • Wolfgang MulzerEmail author
  • Liam Roditty
  • Paul Seiferth
Article
  • 7 Downloads

Abstract

Let \(P \subset \mathbb {R}^d\) be a set of n points in d dimensions such that each point \(p \in P\) has an associated radius\(r_p > 0\). The transmission graphG for P is the directed graph with vertex set P such that there is an edge from p to q if and only if \(|pq| \le r_p\), for any \(p, q \in P\). A reachability oracle is a data structure that decides for any two vertices \(p, q \in G\) whether G has a path from p to q. The quality of the oracle is measured by the space requirement S(n), the query time Q(n), and the preprocessing time. For transmission graphs of one-dimensional point sets, we can construct in \(O(n \log n)\) time an oracle with \(Q(n) = O(1)\) and \(S(n) = O(n)\). For planar point sets, the ratio \(\Psi \) between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on \(\Psi \): the first works only for \(\Psi < \sqrt{3}\) and achieves \(Q(n) = O(1)\) with \(S(n) = O(n)\) and preprocessing time \(O(n\log n)\); the second data structure gives \(Q(n) = O(\Psi ^3 \sqrt{n})\) and \(S(n) = O(\Psi ^3 n^{3/2})\); the third data structure is randomized with \(Q(n) = O(n^{2/3}\log ^{1/3} \Psi \log ^{2/3} n)\) and \(S(n) = O(n^{5/3}\log ^{1/3} \Psi \log ^{2/3} n)\) and answers queries correctly with high probability.

Notes

Acknowledgements

We like to thank Günter Rote and the anonymous reviewers for valuable comments, in particular for pointing out a drastic simplification for the one-dimensional reachability oracle.

References

  1. 1.
    Alber, J., Fiala, J.: Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms 52(2), 134–151 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arikati, S., Chen, D. Z., Chew, L. P., Das, G., Smid, M., Zaroliagis, C. D.: Planar spanners and approximate shortest path queries among obstacles in the plane. In: Proceedings of 4th Annual European Symposium Algorithms (ESA), pp. 514–528 (1996)CrossRefGoogle Scholar
  3. 3.
    Chen, D. Z., Xu, J.: Shortest path queries in planar graphs. In: Proceedings of 32nd Annual ACM Symposium on Theory of Computing (STOC), pp. 469–478 (2000)Google Scholar
  4. 4.
    Cohen, E., Halperin, E., Kaplan, H., Zwick, U.: Reachability and distance queries via 2-hop labels. SIAM J. Comput. 32(5), 1338–1355 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M .H.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)CrossRefGoogle Scholar
  6. 6.
    Djidjev, H. N.: Efficient algorithms for shortest path queries in planar digraphs. In: Proceedings of 22nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 151–165 (1996)CrossRefGoogle Scholar
  7. 7.
    Federickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Holm, J., Rotenberg, E., Thorup, M.: Planar reachability in linear space and constant time. In: Proceedings 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 370–389 (2015)Google Scholar
  9. 9.
    Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P.: Spanners and reachability oracles for directed transmission graphs. In: Proceedings of 31st International Symposium on Computational Geometry (SoCG), pp. 156–170 (2015)Google Scholar
  10. 10.
    Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P.: Spanners for directed transmission graphs. arXiv:1601.07798 (2016)Google Scholar
  11. 11.
    Peleg, D., Roditty, L.: Localized spanner construction for ad hoc networks with variable transmission range. ACM Trans. Sens. Netw. 7(3), 25:1–25:14 (2010)CrossRefGoogle Scholar
  12. 12.
    Pǎtraşcu, M.: Unifying the landscape of cell-probe lower bounds. SIAM J. Comput. 40(3), 827–847 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. J. ACM 51(6), 993–1024 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    von Rickenbach, P., Wattenhofer, R., Zollinger, A.: Algorithmic models of interference in wireless ad hoc and sensor networks. IEEE/ACM Trans. Netw. 17(1), 172–185 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.Department of Computer ScienceBar Ilan UniversityRamat GanIsrael

Personalised recommendations