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Wireless Networks

, Volume 16, Issue 1, pp 227–235 | Cite as

On routing with guaranteed delivery in three-dimensional ad hoc wireless networks

  • Stephane DurocherEmail author
  • David Kirkpatrick
  • Lata Narayanan
Article

Abstract

We study the problem of routing in three-dimensional ad hoc networks. We are interested in routing algorithms that guarantee delivery and are k-local, i.e., each intermediate node v’s routing decision only depends on knowledge of the labels of the source and destination nodes, of the subgraph induced by nodes within distance k of v, and of the neighbour of v from which the message was received. We model a three-dimensional ad hoc network by a unit ball graph, where nodes are points in three-dimensional space, and for each node v, there is an edge between v and every node u contained in the unit-radius ball centred at v. The question of whether there is a simple local routing algorithm that guarantees delivery in unit ball graphs has been open for some time. In this paper, we answer this question in the negative: we show that for any fixed k, there can be no k-local routing algorithm that guarantees delivery on all unit ball graphs. This result is in contrast with the two-dimensional case, where 1-local routing algorithms that guarantee delivery are known. Specifically, we show that guaranteed delivery is possible if the nodes of the unit ball graph are contained in a slab of thickness \(1/\sqrt{2}.\) However, there is no k-local routing algorithm that guarantees delivery for the class of unit ball graphs contained in thicker slabs, i.e., slabs of thickness \(1/\sqrt{2} + \epsilon\) for some \( \epsilon > 0.\) The algorithm for routing in thin slabs derives from a transformation of unit ball graphs contained in thin slabs into quasi unit disc graphs, which yields a 2-local routing algorithm. We also show several results that further elaborate on the relationship between these two classes of graphs.

Keywords

Ad hoc networks Unit ball graph Routing Distributed algorithms Quasi unit disc graph 

Notes

Acknowledgements

The authors would like to thank Prosenjit Bose for providing helpful answers to our questions regarding hardness results for memoryless routing algorithms as well as Stephen Wismath, Ethan Kim, and John Iacono for discussing ideas which resulted in a preliminary algorithm for routing in unit ball graphs at the 2007 Bellairs Workshop on Computational Geometry.

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Stephane Durocher
    • 1
    Email author
  • David Kirkpatrick
    • 2
  • Lata Narayanan
    • 3
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  3. 3.Department of Computer ScienceConcordia UniversityMontrealCanada

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