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- 5log 2nlog 2D) bits, where D is the (Euclidean) diameter of UD (S). The header size is O(log nlog D) bits.
- Routing scheme
- Unit disk graph
- Well-separated pair decomposition