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posted on 01.01.1975by Michael I. Shamos, Dan Hoey
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.