Title

Closest-point Problems

Date of Original Version

1975

Type

Working Paper

Rights Management

All Rights Reserved

Abstract or Description

A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2 ) or worse. The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space. The Voronoi diagram is used to obtain D(N log N) algorithms for all of the problems.