Voronoi Diagrams with a Transportation Network on the Euclidean Plane

This paper investigates geometric and algorithmic properties of the Voronoi diagram with a transportation network on the Euclidean plane. With a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm + m + nm log n) algorithm to compute the Voronoi diagram with a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needsO(nm+m +n log n) time withO(m(n+m)) space and the other O(nm logn + m logm) time with O(n + m) space. Both algorithms output the diagram with linear size.