Routing in Unit Disk Graphs

Let S⊂ R² be a set of n sites. The unit disk graph UD (S) on S has vertex set S and an edge between two distinct sites s, t∈ S if and only if s and t have Euclidean distance | st| ≤ 1. A routing scheme R for UD (S) assigns to each site s∈ S a labelℓ(s) and a routing tableρ(s). For any two sites s, t∈ 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 labelℓ(t) of the target, the scheme R consults the routing table ρ(r) to compute a neighbor r^′ of r, a new header h^′, and the label ℓ(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 UD (S) , over all pairs of distinct sites in S. For any given ε> 0 , we show how to construct a routing scheme for UD (S) with stretch 1 + ε using labels of O(log n) bits and routing tables of O(ε^{- 5}log ²nlog ²D) bits, where D is the (Euclidean) diameter of UD (S). The header size is O(log nlog D) bits.