A compact piecewise-linear Voronoi diagram for convex sites in the plane

Abstract

In the plane, the post-office problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a compact representation of the Voronoi diagram, using O(k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons with n total vertices, we compute this diagram optimally in O(klog n) deterministic time for the Euclidean metric and in O(klog nlog m) deterministic time for the convex distance function defined by a convex m-gon.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>