### 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) |
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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 |

### 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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## Cite this

*Information Processing Letters*,

*64*(6), 309-314.