Euclidean minimum spanning trees and bichromatic closest pairs

Abstract

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+?), for some fixed ?>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)+?+m logn+n logm) for the bichromatic closest pair problem andO(N2?2/([d/2]+1)?) for the Euclidean minimum spanning tree problem, for any positive ?.