Outing S Chemes for D Isk G Raphs and P Olygons

Routing in networks is a fundamental problem that occurred in the 1980’s and was well studied since then. However, there are still open problems, which have not been solved until now. In this thesis we propose routing schemes for special graph classes. Let G = (V,E) be a weighted network or graph. For any two nodes p, q ∈ V I would like to be able to route a data package from p to q. A routing schemeR assigns to each node p ∈ V a label l(p) ∈ {0,1}∗ and a routing table ρ(p) ∈ {0,1}∗. The label identifies the node in the network and the routing table is its own local read-only memory. Now the scheme works in the following way: the scheme starts with a current node p, the label of a target node and some additional information in the data package, called header. As a next step, the scheme computes a new node in the network, where the data package is forwarded to. It can use the information in the header and the local memory. The resulting sequence of nodes is the routing path. The stretch ofR is the maximum ratio of the Euclidean length of the routing path and the shortest path. Usually, wireless networks are modelled using (unit) disk graphs. A disk graph DG(S) consists of a set S of n disks with certain radii. The centres of the disks correspond to transmitters of the wireless network. We say that two transmitters are connected, iff their disks intersect. From a practical point of view it is reasonable to say that the scope of the transmitters does not differ too much. Now for any given > 0, we show how to find a routing scheme for disk graphs with n sites, constant bounded radius ratio and diameter D. This routing scheme has stretch 1 + and uses O(logn) bits for the labels. The header size is O(logn logD) and the routing tables need O( −6 log n logD) bits. The preprocessing time is O(n logn + −4n2 + −6n logn logD). Travelling in polygons is a well-known problem as well. The visibility graph VG(P ) of a polygon P with n vertices and h holes is a graph in which two vertices in P are connected, iff they can see each other, meaning that the line segment between the two vertices is contained in P . We present the first routing scheme for polygons with holes. For any > 0 the routing scheme provides the stretch 1 + and uses no additional information in the header of a data package. The labels have O(logn) bits and the corresponding routing tables are of size O( −1h logn). The preprocessing time is O(n + nh −1) and can be improved for simple polygons to O(n + n −1).