The complexity of finding minimal Voronoi covers with applications to machine learning

Abstract

Our goal in this paper is to examine the application of Voronoi diagrams, a fundamental concept of computational geometry, to the nearest neighbor algorithm used in machine learning. We consider the question “Given a planar polygonal tessellation T and an integer k, is there a set of k points whose Voronoi diagram contains every edge in T?” We show that this question is NP-hard. We encountered this problem while studying a learning model in which we seek the minimum sized set of training examples needed to teach a given geometric concept to a nearest neighbor learning program. That is, given a concept which can be described by a planar tessellation, we are seeking to construct the smallest set of points whose Voronoi diagram is consistent with the given tessellation. In a sense, this question captures the difficulty of teaching the nearest neighbor algorithm a simple structure, using a minimal number of examples. In addition, we consider the natural inverse to the problem of computing Voronoi diagrams. Given a planar polygonal tessellation T, we describe an algorithm to find a set of points whose Voronoi diagram is T, if such a set exists.