Higher Order City Voronoi Diagrams

TL;DR

This paper studies the structural complexity and algorithms for higher-order Voronoi diagrams in city metrics, considering transportation networks, and provides bounds and efficient computation methods.

Contribution

It introduces bounds on the complexity of higher-order city Voronoi diagrams and develops algorithms for their computation considering transportation networks.

Findings

Complexity bounds differ from Euclidean case due to transportation network.

Developed an O(k^2(n + c) log n) algorithm for kth-order diagrams.

Created an O(nc log^2(n + c) log n) algorithm for farthest-site diagrams.

Abstract

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 {\Omega}(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 {\Theta}(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.