Separating and shattering long line segments

  • Alon Efrat
  • Otfried Schwarzkopf
Session 3a: Invited Presentation
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1178)


A line l is called a separator for a set S of objects in the plane if l avoids all the objects and partitions S into two non-empty subsets, lying on both sides of l. A set L of lines is said to shatter S if each line of L is a separator for S, and every two objects of S are separated by at least one line of L. We give a simple algorithm to construct the set of all separators for a given set S of n line segments in time O(n log n), provided the ratio between the diameter of S and the length of the shortest line segment is bounded by a constant. We also give an O(n log n)-time algorithm to determine a set of lines shattering S, improving (for this setting) the O(n2 log n) time algorithm of Freimer, Mitchell and Piatko.


Computational Geometry BSP-trees line-seperation 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Alon Efrat
    • 1
  • Otfried Schwarzkopf
    • 2
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel-AvivIsrael
  2. 2.Dept. of Computer SciencePohang University of Science and TechnologyPohangSouth Korea

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