Facility Location on a Polyhedral Surface

Given an orientable genus-0 polyhedral surface defined by n triangles, and a set of m point sites} on it, we would like to identify its 1-center, i.e., the location on the surface that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the surface. To compute the 1-center, we compute the furthest-site Voronoi diagram of the sites on the polyhedral surface. We show that the diagram has maximum combinatorial complexity (m n ²), and present an algorithm that computes the diagram in O(m n ²log mlog n) expected time. The 1-center can then be identified in time linear in the size of the diagram.