Farthest neighbors and center points in the presence of rectangular obstacles

We study several natural proximity and facility location problems that arise for a set P of n points and a set R of m disjoint rectangular obstacles in the plane, where distances are measured according to the L₁ shortest path (geodesic) metric. In particular, we compute, in time O(mn log(m + n)), a data structure of size O(mn) that supports O(log(m + n))-time farthest point queries; we avoid computing the more complicated farthest neighbor Voronoi diagram, whose combinatorial complexity we show to be Θ(mn). We study the center point problem, finding in O(mn log(m + n)) time a center point (and the set of center points) that minimize the maximum distance to sites of P; this result improves the best previous bound by a factor of roughly m. In addition, we give algorithms for approximating the diameter, D, and radius, r, of P, including methods to (i) compute a pair of points a, b ∈ P, such that d(a,b) ≥ (1 - ε)D, in O(n log n + 1/ε(n + m)log m) time; and (ii) compute a po int c′, such that max {d(p,c′) | p ∈ P} ≥ (1 + ε)r, in O(n log(m + n) + (m/ε)log(m + 1/ε)) time. Finally, we show that for all the problems above it is enough to consider only a subset of P. This subset is likely to be much smaller than P, it is computable in O(n log n) time, and using it results in significantly decreased runtime in practice.