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 2m + n log 2n) for computing bi-chromatic farthest neighbors between n red points and m blue points in E 3. The algorithm can also be used to compute all farthest neighbors or external farthest neighbors of n points in E 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 E 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 ε and run in time O(nε (1−k) 2 log n) or O(nε (1−k) 2 log 2n) in k-dimensional space.