Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inEd in timeO(Fd(N,N) logdN), whereFd(n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inEd. IfFd(N,N)=Ω(N1+ε), for some fixed ɛ>0, then the running time improves toO(Fd(N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)2/3+m log2n+n log2m) inE3, which yields anO(N4/3 log4/3N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE3. Ind≥4 dimensions we obtain expected timeO((nm)1−1/([d/2]+1)+ε+m logn+n logm) for the bichromatic closest pair problem andO(N2−2/([d/2]+1)ε) for the Euclidean minimum spanning tree problem, for any positive ɛ.

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