OPTIMAL CONSTRUCTION OF THE CITY VORONOI DIAGRAM

We address proximity problems in the presence of roads on the L1 plane. More specifically, we present the first optimal algorithm for constructing the city Voronoi diagram. We apply the continuous Dijkstra method to obtain an optimal algorithm for building a shortest path map for a given source, and then it extends to that for the city Voronoi diagram. Moreover, our algorithm can be extended to other generalized situations including metric spaces induced by roads and obstacles together.

SHORTEST PATHS AMONG OBSTACLES IN THE PLANE

We give a subquadratic (O(n3/2+∊) time and O(n) space) algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles; previous time bounds were at least quadratic in n, in the worst case. The method avoids use of visibility graphs, relying instead on the continuous Dijkstra paradigm. The output is a shortest path map (of size O(n)) with respect to a given source point, which allows shortest path length queries to be answered in time O( log n). The algorithm extends to the case of multiple source points, yielding a method to compute a Voronoi diagram with respect to the shortest path metric.

SHORTEST PATHS ON A POLYHEDRON, Part I: COMPUTING SHORTEST PATHS

We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for processing queries of shortest paths between the source point and any destination point. Our algorithm uses a new approach which deviates from the conventional “continuous Dijkstra” technique. Our algorithm has time complexity O(n2) and space complexity Θ(n).