### 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(n^{2} log n) time algorithm of Freimer, Mitchell and Piatko.

Original language | English (US) |
---|---|

Pages (from-to) | 309-314 |

Number of pages | 6 |

Journal | Information Processing Letters |

Volume | 64 |

Issue number | 6 |

State | Published - Dec 29 1997 |

Externally published | Yes |

### Fingerprint

### Keywords

- BSP-trees
- Computational geometry
- Line separation

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Information Processing Letters*,

*64*(6), 309-314.

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

Research output: Contribution to journal › Article

*Information Processing Letters*, vol. 64, no. 6, pp. 309-314.

}

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 -