Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m < n/polylog n. Moreover, the algorithms for the nearest-point and farthest-point Voronoi diagrams are optimal for m < n/polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.