Routing in Unit Disk Graphs

Abstract

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