Routing from Pentagon to Octagon Delaunay Graphs

Abstract

The standard Delaunay triangulation is a geometric graph whose vertices are points in the plane, and two vertices share an edge if they lie on the boundary of an empty disk. If the disk is replaced with a homothet of a fixed convex shape C, then the resulting graph is called a C-Delaunay graph. We study the problem of local routing in C-Delaunay graphs where C is a regular polygon having five to eight sides. In particular, we generalize the routing algorithm of Chew for square-Delaunay graphs (Chew. SCG 1986, 169-177) in order to obtain the following approximate upper bounds of 4.640, 6.429, 8.531 and 4.054 on the spanning and routing ratios for pentagon-, hexagon-, septagon-, and octagon-Delaunay graphs, respectively. The exact expression for the upper bounds of the routing ratio is Ψ(n):= √{1+((cos(2π/n)+n-1)/sin(2π/n))^2} (if n ∈ {5,6,7}), √{1+((cos(π/8)cos(3π/8)+3)/(cos(π/8)sin(3π/8)))^2} (if n = 8).
We show that these bounds are tight for the output of our routing algorithm by providing a point set where these bounds are achieved. We also include lower bounds of 1.708 and 1.995 on the spanning and routing ratios of the pentagon-Delaunay graph.
Our upper bounds yield a significant improvement over the previous routing ratio upper bounds for this problem, which previously sat at around 400 for the pentagon, septagon, and octagon as well as 18 for the hexagon. Our routing ratios also provide significant improvements over the previously best known spanning ratios for pentagon-, septagon- and octagon-Delaunay graphs, which were around 45.