Separating and shattering long line segments

Alon Efrat, Otfried Schwarzkopf

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A line I is called a separator for a set S of objects in the plane if I 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.

Original languageEnglish (US)
Title of host publicationAlgorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings
PublisherSpringer Verlag
Pages36-44
Number of pages9
Volume1178
ISBN (Print)3540620486, 9783540620488
StatePublished - 1996
Externally publishedYes
Event7th International Symposium on Algorithms and Computation, ISAAC 1996 - Osaka, Japan
Duration: Dec 16 1996Dec 18 1996

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1178
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other7th International Symposium on Algorithms and Computation, ISAAC 1996
CountryJapan
CityOsaka
Period12/16/9612/18/96

Fingerprint

Line segment
Separators
Separator
Line
Partition
Subset
Object

Keywords

  • BSP-trees
  • Computational geometry
  • Line-separation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Efrat, A., & Schwarzkopf, O. (1996). Separating and shattering long line segments. In Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings (Vol. 1178, pp. 36-44). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1178). Springer Verlag.

Separating and shattering long line segments. / Efrat, Alon; Schwarzkopf, Otfried.

Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings. Vol. 1178 Springer Verlag, 1996. p. 36-44 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1178).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Efrat, A & Schwarzkopf, O 1996, Separating and shattering long line segments. in Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings. vol. 1178, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1178, Springer Verlag, pp. 36-44, 7th International Symposium on Algorithms and Computation, ISAAC 1996, Osaka, Japan, 12/16/96.
Efrat A, Schwarzkopf O. Separating and shattering long line segments. In Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings. Vol. 1178. Springer Verlag. 1996. p. 36-44. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Efrat, Alon ; Schwarzkopf, Otfried. / Separating and shattering long line segments. Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings. Vol. 1178 Springer Verlag, 1996. pp. 36-44 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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