Separating and shattering long line segments

Alon Efrat, Otfried Schwarzkopf

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

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 randomized algorithm to construct the set of all separators for a given set S of n line segments in expected 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 a randomized algorithm to determine a set of lines shattering S, whose expected running time is O(n log n), improving (for this setting) the (deterministic) O(n2 log n) time algorithm of Freimer, Mitchell and Piatko.

Original languageEnglish (US)
Pages (from-to)309-314
Number of pages6
JournalInformation Processing Letters
Volume64
Issue number6
StatePublished - Dec 29 1997
Externally publishedYes

Keywords

  • BSP-trees
  • Computational geometry
  • Line separation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Information Systems
  • Computer Science Applications

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