The Geodesic Farthest-Point Voronoi Diagram in a Simple Polygon

Given a set of point sites in a simple polygon, the geodesic farthest-point Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an \(O(n\log \log n+ m\log m)\)-time algorithm to compute the geodesic farthest-point Voronoi diagram of m point sites in a simple n-gon. This improves the previously best known algorithm by Aronov et al. (Discrete Comput Geom 9(3):217–255, 1993). In the case that all point sites are on the boundary of the simple polygon, we can compute the geodesic farthest-point Voronoi diagram in \(O((n+m) \log \log n)\) time.