Half-Plane Voronoi Diagram

In normal Voronoi diagram, each site is able to see all points in the plane. In this paper, we study the problem such that each site is only able to see half-plane and construct the so-called Half-plane Voronoi Diagram (HPVD). We show that the half-plane Voronoi cell of each site is not necessary convex and it could consist of many disjoint regions. We prove that the complexity of the HPVD of n sites is $O(n^2)$. Then we give an algorithm of $O(n\log n)$ time and $O(n)$ space to construct HPVD such that the boundary lines of all half-planes are parallel and the visible half-planes are in the same side of the corresponding boundary lines. For the parallel boundary lines where the visible half-planes could be in either side of the corresponding boundary lines, we give an algorithm of $O(n^2)$ time and $O(n^2)$ space to construct HPVD.