, Volume 9, Issue 5, pp 495–514 | Cite as

Selecting distances in the plane

  • Pankaj K. Agarwal
  • Boris Aronov
  • Micha Sharir
  • Subhash Suri


We present a randomized algorithm for computing the kth smallest distance in a set ofn points in the plane, based on the parametric search technique of Megiddo [Mel]. The expected running time of our algorithm is O(n4/3 log8/3n). The algorithm can also be made deterministic, using a more complicated technique, with only a slight increase in its running time. A much simpler deterministic version of our procedure runs in time O(n3/2 log5/2n). All versions improve the previously best-known upper bound ofO(@#@ n9/5 log4/5n) by Chazelle [Ch]. A simpleO(n logn)-time algorithm for computing an approximation of the median distance is also presented.

Key words

Parametric search Arrangements Random-sampling 


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Copyright information

© Springer-Verlag New York Inc 1993

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Boris Aronov
    • 2
  • Micha Sharir
    • 3
    • 4
  • Subhash Suri
    • 5
  1. 1.Computer Science DepartmentDuke UniversityDurhamUSA
  2. 2.Department of Computer SciencePolytechnic UniversityBrooklynUSA
  3. 3.Department of Computer Science, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  5. 5.BellcoreMorristownUSA

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