# Euclidean minimum spanning trees and bichromatic closest pairs

Agarwal P, Edelsbrunner H, Schwarzkopf O, Welzl E. 1991. Euclidean minimum spanning trees and bichromatic closest pairs. Discrete & Computational Geometry. 6(1), 407–422.

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Author

Agarwal, Pankaj K;
Edelsbrunner, Herbert

^{IST Austria}^{}; Schwarzkopf, Otfried ; Welzl, EmoAbstract

We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of N points in Ed in time O(Fd (N,N) logd N), where Fd (n,m) is the time required to compute a bichromatic closest pair among n red and m green points in Ed . If Fd (N,N)=Ω(N1+ε), for some fixed e{open}>0, then the running time improves to O(Fd (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 E3, which yields an O(N4/3 log4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree of N points in E3. 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(N2-2/([d/2]+1)ε) for the Euclidean minimum spanning tree problem, for any positive e{open}.

Publishing Year

Date Published

1991-12-01

Journal Title

Discrete & Computational Geometry

Acknowledgement

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648, National Science Foundation under Grant CCR-8714565.

Volume

6

Issue

1

Page

407 - 422

IST-REx-ID

### Cite this

Agarwal P, Edelsbrunner H, Schwarzkopf O, Welzl E. Euclidean minimum spanning trees and bichromatic closest pairs.

*Discrete & Computational Geometry*. 1991;6(1):407-422. doi:10.1007/BF02574698Agarwal, P., Edelsbrunner, H., Schwarzkopf, O., & Welzl, E. (1991). Euclidean minimum spanning trees and bichromatic closest pairs.

*Discrete & Computational Geometry*. Springer. https://doi.org/10.1007/BF02574698Agarwal, Pankaj, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. “Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs.”

*Discrete & Computational Geometry*. Springer, 1991. https://doi.org/10.1007/BF02574698.P. Agarwal, H. Edelsbrunner, O. Schwarzkopf, and E. Welzl, “Euclidean minimum spanning trees and bichromatic closest pairs,”

*Discrete & Computational Geometry*, vol. 6, no. 1. Springer, pp. 407–422, 1991.Agarwal, Pankaj, et al. “Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs.”

*Discrete & Computational Geometry*, vol. 6, no. 1, Springer, 1991, pp. 407–22, doi:10.1007/BF02574698.