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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 kth-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. 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^2(n + c) log n)-time iterative algorithm to compute the kth-order city Voronoi diagram and an O(nc log^2(n + c) log n)-time divide-and-conquer algorithm to compute the farthest-site city Voronoi diagram.