On Routing with Guaranteed Delivery in Three-Dimensional Ad Hoc Wireless Networks

  • Stephane Durocher
  • David Kirkpatrick
  • Lata Narayanan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4904)


We study routing algorithms for three-dimensional ad hoc networks 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 ℝ3, and nodes u and v are joined by an edge if and only if the distance between u and v is at most one.

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 ε> 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.


Destination Node Label Graph Correspondent Node Geometric Graph Thin Slab 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Stephane Durocher
    • 1
  • David Kirkpatrick
    • 2
  • Lata Narayanan
    • 3
  1. 1.School of Computer ScienceMcGill UniversityCanada
  2. 2.Department of Computer ScienceUniversity of British ColumbiaCanada
  3. 3.Department of Computer ScienceConcordia UniversityCanada

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