# Farthest neighbors, maximum spanning trees and related problems in higher dimensions

## Abstract

We present a randomized algorithm of expected time complexity *O*(*m*^{2/3}*n*^{2/3}log^{4/3}*m*+*m*log^{2}*m*+*n*log^{2}*n*) for computing bi-chromatic farthest neighbors between *n* red points and *m* blue points in ɛ^{3}. The algorithm can also be used to compute all farthest neighbors or external farthest neighbors of *n* points in ɛ^{3} in *O*(*n*^{4/3}log^{4/3}*n*) expected time. Using these procedures as building blocks, we can compute a Euclidean maximum spanning tree or a minimum-diameter two-partition of *n* points in ɛ^{3} in *O*(*n*^{4/3}log^{7/3}*n*) expected time. The previous best bound for these problems was *O*(*n*^{3/2}log^{1/2}*n*). Our algorithms can be extended to higher dimensions.

We also propose fast and simple approximation algorithms for these problems. These approximation algorithms produce solutions that approximate the true value with a relative accuracy ɛ and run in time *O*(*n*ɛ^{(1−k)/2}log*n*) or *O*(*n*ɛ^{(1−k)/2}log^{2}*n*) in *k*-dimensional space.

## Keywords

Span Tree Minimum Span Tree Voronoi Diagram Computational Geometry Blue Point## Preview

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