Competitive Local Routing with Constraints

Let P be a set of n points in the plane and S a set of non-crossing line segments between vertices in P , called constraints. Two vertices are visible if the straight line segment connecting them does not properly intersect any constraints. The constrained θm-graph is constructed by partitioning the plane around each vertex into m disjoint cones, each with aperture θ = 2π/m, and adding an edge to the ‘closest’ visible vertex in each cone. We consider how to route on the constrained θ6graph. We first show that no deterministic 1-local routing algorithm is o( √ n)-competitive on all pairs of vertices of the constrained θ6-graph. After that, we show how to route between any two visible vertices of the constrained θ6-graph using only 1-local information. Our routing algorithm guarantees that the returned path has length at most 2 times the Euclidean distance between the source and destination. Additionally, we provide a 1-local 18-competitive routing algorithm for visible vertices in the constrained half-θ6-graph, a subgraph of the constrained θ6-graph that is equivalent to the Delaunay graph where the empty region is an equilateral triangle. To the best of our knowledge, these are the first local routing algorithms in the constrained setting with guarantees on the length of the returned path. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory