, Volume 6, Issue 1, pp 407-422

Euclidean minimum spanning trees and bichromatic closest pairs

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Abstract

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN 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 amongn red andm 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 logn logm)2/3+m log2 n+n log2 m) inE 3, which yields anO(N 4/3 log4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE 3. 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(N 2−2/([d/2]+1)ε) for the Euclidean minimum spanning tree problem, for any positive ɛ.

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).