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

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

Title of host publication | Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings |

Publisher | Springer Verlag |

Pages | 36-44 |

Number of pages | 9 |

Volume | 1178 |

ISBN (Print) | 3540620486, 9783540620488 |

State | Published - 1996 |

Externally published | Yes |

Event | 7th International Symposium on Algorithms and Computation, ISAAC 1996 - Osaka, Japan Duration: Dec 16 1996 → Dec 18 1996 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 1178 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 7th International Symposium on Algorithms and Computation, ISAAC 1996 |
---|---|

Country | Japan |

City | Osaka |

Period | 12/16/96 → 12/18/96 |

### Fingerprint

### Keywords

- BSP-trees
- Computational geometry
- Line-separation

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Separating and shattering long line segments

AU - Efrat, Alon

AU - Schwarzkopf, Otfried

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

KW - BSP-trees

KW - Computational geometry

KW - Line-separation

UR - http://www.scopus.com/inward/record.url?scp=84990231580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990231580&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84990231580

SN - 3540620486

SN - 9783540620488

VL - 1178

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 36

EP - 44

BT - Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings

PB - Springer Verlag

ER -