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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
Article

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(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.

Keywords

Minimum Span Tree Voronoi Diagram Computational Geometry Voronoi Cell Query Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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