# Euclidean minimum spanning trees and bichromatic closest pairs

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DOI: 10.1007/BF02574698

- Cite this article as:
- Agarwal, P.K., Edelsbrunner, H., Schwarzkopf, O. et al. Discrete Comput Geom (1991) 6: 407. doi:10.1007/BF02574698

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## Abstract

We present an algorithm to compute a Euclidean minimum spanning tree of a given set*S* of*N* points inE^{d} in timeO(F_{d}(*N,N*) log^{d}*N*), whereF_{d}(*n,m*) is the time required to compute a bichromatic closest pair among*n* red and*m* green points inE^{d}. IfF_{d}(*N,N*)=Ω(*N*^{1+ε}), for some fixed ɛ>0, then the running time improves toO(F_{d}(*N,N*)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((*nm* log*n* log*m*)^{2/3}+*m* log^{2}*n*+*n* log^{2}*m*) inE_{3}, which yields anO(*N*^{4/3} log^{4/3}*N*) expected time, algorithm for computing a Euclidean minimum spanning tree of*N* points inE^{3}. In*d*≥4 dimensions we obtain expected timeO((*nm*)^{1−1/([d/2]+1)+ε}+*m* log*n+n* log*m*) for the bichromatic closest pair problem andO(*N*^{2−2/([d/2]+1)ε}) for the Euclidean minimum spanning tree problem, for any positive ɛ.