On Higher Order Voronoi Diagrams of Line Segments

We analyze structural properties of the order-k Voronoi diagram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single order-k Voronoi region may consist of Omega(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of non-intersecting segments remains O(k(n -k)) similarly to points. For intersecting line segments the structural complexity remains O(k(n -k)) for k >= n/2.