Separating and shattering long line segments

Alon Efrat, Otfried Schwarzkopf

Research output: Contribution to journalArticle

2 Citations (Scopus)

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

Fingerprint

Line segment
Separators
Separator
Line
Randomized Algorithms
Partition
Subset
Object

Keywords

  • BSP-trees
  • Computational geometry
  • Line separation

ASJC Scopus subject areas

  • Computational Theory and Mathematics

Cite this

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

In: Information Processing Letters, Vol. 64, No. 6, 29.12.1997, p. 309-314.

Research output: Contribution to journalArticle

Efrat, A & Schwarzkopf, O 1997, 'Separating and shattering long line segments', Information Processing Letters, vol. 64, no. 6, pp. 309-314.
Efrat, Alon ; Schwarzkopf, Otfried. / Separating and shattering long line segments. In: Information Processing Letters. 1997 ; Vol. 64, No. 6. pp. 309-314.
@article{5752cc717152409685ae0181c3aeb022,
title = "Separating and shattering long line segments",
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.",
keywords = "BSP-trees, Computational geometry, Line separation",
author = "Alon Efrat and Otfried Schwarzkopf",
year = "1997",
month = "12",
day = "29",
language = "English (US)",
volume = "64",
pages = "309--314",
journal = "Information Processing Letters",
issn = "0020-0190",
publisher = "Elsevier",
number = "6",

}

TY - JOUR

T1 - Separating and shattering long line segments

AU - Efrat, Alon

AU - Schwarzkopf, Otfried

PY - 1997/12/29

Y1 - 1997/12/29

N2 - 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.

AB - 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.

KW - BSP-trees

KW - Computational geometry

KW - Line separation

UR - http://www.scopus.com/inward/record.url?scp=0042634021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042634021&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0042634021

VL - 64

SP - 309

EP - 314

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 6

ER -