The city Voronoi diagram is induced by quickest paths in the L 1 plane, made faster by an isothetic transportation network. We investigate the rich geometric and algorithmic properties of city Voronoi diagrams, and report on their use in processing quickest-path queries. In doing so, we revisit the fact that not every Voronoi-type diagram has interpretations in both the distance model and the wavefront model. Especially, straight skeletons are a relevant example where an interpretation in the former model is lacking. We clarify the relationship between these models, and further draw a connection to the bisector-defined abstract Voronoi diagram model, with the particular goal of computing the city Voronoi diagram efficiently.
A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H.The highway hull H(S, H) of a point set S is the minimal set containing S as well as the shortest paths between all pairs of points in H(S, H), using the highway time distance.We provide a Θ(n log n) worst-case time algorithm to find the highway hull under the L1 metric, as well as an O(n log 2 n) time algorithm for the L2 metric which improves the best known result of O(n 2 ) [15,20]. We also define and construct the useful region of the plane: the region that a highway must intersect in order that the shortest path between at least one pair of points uses the highway.
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