Minimum Dilation Triangulations

Given a planar graph G, the graph theoretic dilation of G is defined as the maximum ratio of the shortest-path distance and the Euclidean distance between any two vertices of G. Given a planar point set S, the graph theoretic dilation of S is the minimum graph theoretic dilation that any triangulation of S can achieve. We study the graph theoretic dilation of the regular n-gon. In particular, we compute a simple lower bound for the graph theoretic dilation of the regular n-gon and use this bound in order to derive an efficient approximation algorithm that computes a triangulation whose graph theoretic dilation is within a factor of 1 +O 1/ √ log n of the optimum. Furthermore, we demonstrate how the general concept of exclusion regions applies to minimum dilation triangulations.