Stephane Durocher

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

We study routing algorithms for three-dimensional ad hoc networks that guarantee delivery and are k-local, i.e., each intermediate node υs 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 υ, and of the neighbour of υ 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 υ 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/√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/√2+Ɛ for some Ɛ > 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.

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