{"date_published":"1991-12-01T00:00:00Z","author":[{"first_name":"Pankaj","full_name":"Agarwal, Pankaj K","last_name":"Agarwal"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Herbert Edelsbrunner"},{"full_name":"Schwarzkopf, Otfried ","first_name":"Otfried","last_name":"Schwarzkopf"},{"full_name":"Welzl, Emo","first_name":"Emo","last_name":"Welzl"}],"doi":"10.1007/BF02574698","date_updated":"2021-01-12T07:54:13Z","quality_controlled":0,"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}."}],"publication":"Discrete & Computational Geometry","extern":1,"date_created":"2018-12-11T12:06:42Z","_id":"4061","title":"Euclidean minimum spanning trees and bichromatic closest pairs","citation":{"short":"P. Agarwal, H. Edelsbrunner, O. Schwarzkopf, E. Welzl, Discrete & Computational Geometry 6 (1991) 407–422.","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.","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","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","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.","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."},"month":"12","publist_id":"2062","page":"407 - 422","day":"01","volume":6,"type":"journal_article","publication_status":"published","year":"1991","intvolume":" 6","publisher":"Springer","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.","status":"public","issue":"1"}