Iterated Nearest Neighbors and Finding Minimal Polytopes

We introduce a new method for nding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, which were based on high order Voronoi diagrams. Our technique allows us for the rst time to e ciently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to nd minimal convex k-vertex polygons and polytopes, and to improve many previous results. We achieve many of our results via a new algorithm for nding rectilinear nearest neighbors in the plane in time O(n logn+ kn). We also demonstrate a related technique for nding minimum area k-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to nd minimum volume and boundary measure sets in arbitrary dimensions.