# Euclidean minimum spanning trees and bichromatic closest pairs

Published

Journal Article

We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of N points in E d in time O(F d (N,N) log d N), where F d (n,m) is the time required to compute a bichromatic closest pair among n red and m green points in E d . If F d (N,N)=Ω(N 1+ε), for some fixed e{open}>0, then the running time improves to O(F d (N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected time O((nm log n log m)2/3+m log2 n+n log2 m) in E 3, which yields an O(N 4/3 log4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree of N points in E 3. In d≥4 dimensions we obtain expected time O((nm)1-1/([d/2]+1)+ε+m log n+n log m) for the bichromatic closest pair problem and O(N 2-2/([d/2]+1)ε) for the Euclidean minimum spanning tree problem, for any positive e{open}. © 1991 Springer-Verlag New York Inc.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Edelsbrunner, H; Schwarzkopf, O; Welzl, E

### Published Date

- December 1, 1991

### Published In

### Volume / Issue

- 6 / 1

### Start / End Page

- 407 - 422

### Electronic International Standard Serial Number (EISSN)

- 1432-0444

### International Standard Serial Number (ISSN)

- 0179-5376

### Digital Object Identifier (DOI)

- 10.1007/BF02574698

### Citation Source

- Scopus