Algorithmica

pp 1–19 | Cite as

Routing in Unit Disk Graphs

  • Haim Kaplan
  • Wolfgang Mulzer
  • Liam Roditty
  • Paul Seiferth
Article
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Part of the following topical collections:
  1. Special Issue on Theoretical Informatics

Abstract

Let \(S \subset \mathbb {R}^2\) be a set of n sites. The unit disk graph \({{\mathrm{UD}}}(S)\) on S has vertex set S and an edge between two distinct sites \(s,t \in S\) if and only if s and t have Euclidean distance \(|st| \le 1\). A routing scheme R for \({{\mathrm{UD}}}(S)\) assigns to each site \(s \in S\) a label\(\ell (s)\) and a routing table\(\rho (s)\). For any two sites \(s, t \in S\), the scheme R must be able to route a packet from s to t in the following way: given a current siter (initially, \(r = s\)), a headerh (initially empty), and the label\(\ell (t)\) of the target, the scheme R consults the routing table \(\rho (r)\) to compute a neighbor \(r'\) of r, a new header \(h'\), and the label \(\ell (t')\) of an intermediate target \(t'\). (The label of the original target may be stored at the header \(h'\).) The packet is then routed to \(r'\), and the procedure is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in \({{\mathrm{UD}}}(S)\), over all pairs of distinct sites in S. For any given \(\varepsilon > 0\), we show how to construct a routing scheme for \({{\mathrm{UD}}}(S)\) with stretch \(1+\varepsilon \) using labels of \(O(\log n)\) bits and routing tables of \(O(\varepsilon ^{-5}\log ^2 n \log ^2 D)\) bits, where D is the (Euclidean) diameter of \({{\mathrm{UD}}}(S)\). The header size is \(O(\log n \log D)\) bits.

Keywords

Routing scheme Unit disk graph Well-separated pair decomposition 

References

  1. 1.
    Bose, P., Morin, P., Stojmenovic, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. Wirel. Netw. 7(6), 609–616 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Callahan, P., Kosaraju, S.: A decomposition of multidimensional point sets with applications to \(k\)-nearest-neighbors and \(n\)-body potential fields. J. ACM 42(1), 67–90 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chechik, S.: Compact routing schemes with improved stretch. In: Proceedings of the 32nd ACM Symposium on Principles of Distributed Computing (PODC), pp. 33–41 (2013)Google Scholar
  4. 4.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)MATHGoogle Scholar
  6. 6.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry Algorithms and Applications, 3rd edn. Springer, Berlin (2008)MATHGoogle Scholar
  7. 7.
    Fraigniaud, P., Gavoille, C.: Routing in trees. In: Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP), pp. 757–772 (2001)Google Scholar
  8. 8.
    Gao, J., Zhang, L.: Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM J. Comput. 35(1), 151–169 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Giordano, S., Stojmenovic, I.: Position based routing algorithms for ad hoc networks: a taxonomy. In: Cheng, X., Huang, X., Du, D.-Z. (eds.) Ad Hoc Wireless Networking, pp. 103–136. Springer, New York (2004)Google Scholar
  10. 10.
    Gupta, A., Kumar, A., Rastogi, R.: Traveling with a Pez dispenser (or, routing issues in MPLS). SIAM J. Comput. 34(2), 453–474 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Karp, B., Kung, H.T.: GPSR: greedy perimeter stateless routing for wireless networks. In: Proceedings of the 6th Annual International Conference on Mobile Computing and Networking (MOBICOM), pp. 243–254 (2000)Google Scholar
  12. 12.
    Kuhn, F., Wattenhofer, R., Zhang, Y., Zollinger, A.: Geometric ad-hoc routing: of theory and practice. In: Proceedings of the 22nd ACM Symposium on Principles of Distributed Computing (PODC), pp. 63–72 (2003)Google Scholar
  13. 13.
    Mitchell, J.S.B., Mulzer, W.: Proximity algorithms. In: Toth, C., O’Rourke, J., Goodman, J. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, Boca Raton (2017). (to appear)Google Scholar
  14. 14.
    Narasimhan, G., Smid, M.H.M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  15. 15.
    Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM 36(3), 510–530 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Roditty, L., Tov, R.: New routing techniques and their applications. In: Proceedings of the 34th ACM Symposium on Principles of Distributed Computing (PODC), pp. 23–32 (2015)Google Scholar
  17. 17.
    Santoro, N., Khatib, R.: Labelling and implicit routing in networks. Comput. J. 28(1), 5–8 (1985)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. J. ACM 51(6), 993–1024 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Thorup, M., Zwick, U.: Compact routing schemes. In: Proceedings of the 13th ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 1–10 (2001)Google Scholar
  20. 20.
    Yan, C., Xiang, Y., Dragan, F.F.: Compact and low delay routing labeling scheme for unit disk graphs. Comput. Geom. 45(7), 305–325 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

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

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