We investigate the higher-order Voronoi diagrams of n point sites with respect to the geodesic distance in a simple polygon with h > 0 polygonal holes and c corners. Given a set of n point sites, the k~(th)-order Voronoi diagram partitions the plane into several regions such that all points in a region share the same k nearest sites. The nearest-site (first-order) geodesic Voronoi diagram has already been well-studied, and its total complexity is O(n+c). On the other hand, Bae and Chwa recently proved that the total complexity of the farthest-site ((n-1)~(st)-order) geodesic Voronoi diagram and the number of faces in the diagram are Θ(nc) and Θ(nh), respectively. It is of high interest to know what happens between the first-order and the (n-1) ~(st)-order geodesic Voronoi diagrams. In this paper we prove that the total complexity of the kth-order geodesic Voronoi diagram is Θ(k(n - k) + kc), and the number of faces in the diagram is Θ(k(n-k)+kh). Our results successfully explain the variation from the nearest-site to the farthest-site geodesic Voronoi diagrams, i.e., from k = 1 to k = n - 1, and also illustrate the formation of a disconnected Voronoi region, which does not occur in many commonly used distance metrics, such as the Euclidean, L_1, and city metrics. We show that the k~(th)-order geodesic Voronoi diagram can be computed in O(k~2(n+c) log(n+c)) time using an iterative algorithm.
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