Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles.

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-theta(6)-graph (the half-theta(6)-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most 5/root 3 approximate to 2.887 times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-theta(6)-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-theta(6)-graph using O(log n) bits per vertex coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1990), ACM, New York, SIAM, Philadelphia, 1990, pp. 138-148], our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of 15/root 3 approximate to 8.660 on two bounded degree subgraphs of the half-theta(6)-graph.