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 located 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^1.5) in O(N^1.5 log² N) time that supports O(N^0.5 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^1.5 log²N) time. This improves the previous O(mnlogN) bound [1].