Question

Construct the Voronoi polygons for a convex pentagon.

Answer 1

Let P be a list of points in the plane such that the points of P taken in order form the vertices of a convex polygon. We introduce a simple, linear expected-time algorithm for finding the Voronoi diagram of the points in P. Unlike previous results on expected-time algorithms for Voronoi diagrams, this method does not require any assumptions about the distribution of points. With minor modifications, this method can be used to design fast algorithms for certain problems involving unrestricted sets of points. For example, fast expected-time algorithms can be designed to delete a point from a Voronoi diagram, to build an order k Voronoi diagram for an arbitrary set of points, and to determine the smallest enclosing circle for points at the vertices of a convex hull.