Nearest neighbors revisited

Abstract

The nearest neighbor graph (NNG), defined for a set of points in Euclidean space, has found many uses in computational geometry and clustering analysis. Yet there seems to be surprisingly little knowledge about some basic properties of this graph. In this talk, we ask some natural questions that are motivated by geometric embedding problems. For example, in the simulation of many-body systems, it is desirable to map a set of particles to a regular data array so as to preserve neighborhood relations, i.e., to minimize the dilation of NNG. We will derive bounds on the dilation by studying the diameter of NNG. Other properties and applications of NNG will also be discussed. (Joint work with Mike Paterson)