Farthest Neighbor Voronoi Diagram in the Presence of Rectangular Obstacles

We propose an implicit representation for the farthest Voronoi Diagram of a set P of n points in the plane lying outside a set R of m disjoint axes-parallel rectangular obstacles. The distances are measured according to the L1 shortest path (geodesic) metric. In particular, we design a data structure of size O(N) in O(N log N) time that supports O(N logN)-time farthest point queries (where N = m + n). We avoid computing the more complicated farthest neighbor Voronoi diagram, whose combinatorial complexity is Θ(mn). This allows one to compute the diameter (and all farthest pairs) of P in O(N log N) time. This improves the previous O(mn logN) bound [1].