A Simple Framework for the Generalized Nearest Neighbor Problem

Abstract

The problem of finding a nearest neighbor from a set of points in R d to a complex query object has attracted considerable attention due to various applications in computational geometry, bio-informatics, information retrieval, etc. We propose a generic method that solves the problem for various classes of query objects and distance functions in a unified way. Moreover, for linear space requirements the method simplifies the known approach based on ray-shooting in the lower envelope of an arrangement.

On Voronoi Diagram in the Line Space and their Generalizations

“…Unlike the previously used approach to studying such diagrams, based on the concept of duality between points and lines in the plane, our ideas extend to the three-dimensional case, in which a Voronoi diagram of a set of points in the space of all planes is examined, and generalize further to higher dimensions. Though these Voronoi structures are unlikely to allow for a more efficient processing of the nearest neighbor queries than by the existing methods (see [3], [6], [8] for two-dimensional case, [7] for three-dimensional case, and a very recent work [5] for a novel general framework), our results may help to develop a better intuition in regard of their properties, and to admire their inherent beauty.…”