Routing in Unit Disk Graphs

Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth

Abstract: Let $$S \subset \mathbb {R}^2$$ S ⊂ R 2 be a set of n sites. The unit disk graph $${{\mathrm{UD}}}(S)$$ UD ( S ) on S has vertex set S and an edge between two distinct sites $$s,t \in S$$ s , t ∈ S if and only if s and t have Euclidean distance $$|st| \le 1$$ | s t | ≤ 1 . A routing scheme R for $${{\mathrm{UD}}}(S)$$ UD ( S ) assigns to each site $$s \in S$$ s ∈ S a label $$\ell (s)$$ ℓ ( s ) and a routing table $$\rho (s)$$ ρ ( s ) . For any two sites $$s, t \in S$$ s , t ∈ S , the scheme R must be able to route a packet from s to t in the following way: given a current site r (initially, $$r = s$$ r = s ), a header h (initially empty), and the label $$\ell (t)$$ ℓ ( t ) of the target, the scheme R consults the routing table $$\rho (r)$$ ρ ( r ) to compute a neighbor $$r'$$ r ′ of r, a new header $$h'$$ h ′ , and the label $$\ell (t')$$ ℓ ( t ′ ) of an intermediate target $$t'$$ t ′ . (The label of the original target may be stored at the header $$h'$$ h ′ .) The packet is then routed to $$r'$$ 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 $${{\mathrm{UD}}}(S)$$ UD ( S ) , over all pairs of distinct sites in S. For any given $$\varepsilon > 0$$ ε > 0 , we show how to construct a routing scheme for $${{\mathrm{UD}}}(S)$$ UD ( S ) with stretch $$1+\varepsilon $$ 1 + ε using labels of $$O(\log n)$$ O ( log n ) bits and routing tables of $$O(\varepsilon ^{-5}\log ^2 n \log ^2 D)$$ O ( ε - 5 log 2 n log 2 D ) bits, where D is the (Euclidean) diameter of $${{\mathrm{UD}}}(S)$$ UD ( S ) . The header size is $$O(\log n \log D)$$ O ( log n log D ) bits.