Higher Order City Voronoi Diagrams

We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n+ kc), where c is the complexity of the transportation network. This is quite different from the bound O(k(n−k)) in the Euclidean metric [11]. For the special case where k = n−1 the complexity in the Euclidean metric is O(n), while that in the city metric is Θ(nc). Furthermore, we develop an O(k(n+ c) logn)-time iterative algorithm to compute the korder city Voronoi diagram and an O(nc log(n+ c) logn)-time divide-and-conquer algorithm to compute the farthest-site city Voronoi diagram.