Discrete & Computational Geometry

, Volume 6, Issue 3, pp 407–422 | Cite as

Euclidean minimum spanning trees and bichromatic closest pairs

  • Pankaj K. Agarwal
  • Herbert Edelsbrunner
  • Otfried Schwarzkopf
  • Emo Welzl


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(N1+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 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.


Minimum Span Tree Voronoi Diagram Computational Geometry Voronoi Cell Query Time 
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Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Herbert Edelsbrunner
    • 2
  • Otfried Schwarzkopf
    • 3
  • Emo Welzl
    • 3
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Department of Computer ScienceUniversity of Illinois atUrbanaUSA
  3. 3.Institut für Informatik, Fachbereich MathematikFreie Universität BerlinBerlin 33Federal Republic of Germany

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