On the geodesic voronoi diagram of point sites in a simple polygon

Given a simple polygon withn sides in the plane and a set ofk point sites in its interior or on the boundary, compute the Voronoi diagram of the set of sites using the internal geodesic distance inside the polygon as the metric. We describe anO((n + k) log(n + k) logn)-time algorithm for solving this problem and sketch a fasterO((n + k) log(n + k)) algorithm for the case when the set of sites includes all reflex vertices of the polygon in question.Work on this paper was performed while the author held an AT&T Bell Laboratories Ph.D. Scholarship at New York University.