---
_id: '4061'
abstract:
- lang: eng
text: 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}.
acknowledgement: The first, second, and fourth authors acknowledge support from the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National
Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The
second author's work was supported by the National Science Foundation under Grant
CCR-8714565. The third author's work was supported by the Deutsche Forschungsgemeinschaft
under Grant A1 253/1-3, Schwerpunktprogramm "Datenstrukturen und effiziente Algorithmen."
The last two authors' work was also partially supported by the ESPRIT II Basic Research
Action of the EC under Contract No. 3075 (project ALCOM).
article_processing_charge: No
article_type: original
author:
- first_name: Pankaj
full_name: Agarwal, Pankaj
last_name: Agarwal
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Otfried
full_name: Schwarzkopf, Otfried
last_name: Schwarzkopf
- first_name: Emo
full_name: Welzl, Emo
last_name: Welzl
citation:
ama: 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/BF02574698
apa: Agarwal, 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/BF02574698
chicago: Agarwal, 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.
ieee: 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.
ista: Agarwal P, Edelsbrunner H, Schwarzkopf O, Welzl E. 1991. Euclidean minimum
spanning trees and bichromatic closest pairs. Discrete & Computational Geometry.
6(1), 407–422.
mla: 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.
short: P. Agarwal, H. Edelsbrunner, O. Schwarzkopf, E. Welzl, Discrete & Computational
Geometry 6 (1991) 407–422.
date_created: 2018-12-11T12:06:42Z
date_published: 1991-12-01T00:00:00Z
date_updated: 2022-02-24T15:06:41Z
day: '01'
doi: 10.1007/BF02574698
extern: '1'
intvolume: ' 6'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://link.springer.com/article/10.1007/BF02574698
month: '12'
oa: 1
oa_version: Published Version
page: 407 - 422
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer
publist_id: '2062'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Euclidean minimum spanning trees and bichromatic closest pairs
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 6
year: '1991'
...