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

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.