### Abstract

We call a line l a separator for a set S of objects in the plane if l avoids all the objects and partitions S into two nonempty subsets, one consisting of objects lying above l and the other of objects lying below l. In this paper we present an O(n log n)-time algorithm for finding a separator line for a set of n segments, provided the ratio between the diameter of the set of segments and the length of the smallest segment is bounded. The general case is an 'n^{2}-hard' problem, in the sense defined in [10] (see also [8]). Our algorithm is based on the recent results of [15], concerning the union of 'fat' triangles, but we also include an analysis which improves the bounds obtained in [15].

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

Pages (from-to) | 277-288 |

Number of pages | 12 |

Journal | Computational Geometry: Theory and Applications |

Volume | 3 |

Issue number | 5 |

DOIs | |

State | Published - 1993 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*3*(5), 277-288. https://doi.org/10.1016/0925-7721(93)90018-2

**On the union of fat wedges and separating a collection of segments by a line.** / Efrat, Alon; Rote, Günter; Sharir, Micha.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 3, no. 5, pp. 277-288. https://doi.org/10.1016/0925-7721(93)90018-2

}

TY - JOUR

T1 - On the union of fat wedges and separating a collection of segments by a line

AU - Efrat, Alon

AU - Rote, Günter

AU - Sharir, Micha

PY - 1993

Y1 - 1993

N2 - We call a line l a separator for a set S of objects in the plane if l avoids all the objects and partitions S into two nonempty subsets, one consisting of objects lying above l and the other of objects lying below l. In this paper we present an O(n log n)-time algorithm for finding a separator line for a set of n segments, provided the ratio between the diameter of the set of segments and the length of the smallest segment is bounded. The general case is an 'n2-hard' problem, in the sense defined in [10] (see also [8]). Our algorithm is based on the recent results of [15], concerning the union of 'fat' triangles, but we also include an analysis which improves the bounds obtained in [15].

AB - We call a line l a separator for a set S of objects in the plane if l avoids all the objects and partitions S into two nonempty subsets, one consisting of objects lying above l and the other of objects lying below l. In this paper we present an O(n log n)-time algorithm for finding a separator line for a set of n segments, provided the ratio between the diameter of the set of segments and the length of the smallest segment is bounded. The general case is an 'n2-hard' problem, in the sense defined in [10] (see also [8]). Our algorithm is based on the recent results of [15], concerning the union of 'fat' triangles, but we also include an analysis which improves the bounds obtained in [15].

UR - http://www.scopus.com/inward/record.url?scp=38248998602&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38248998602&partnerID=8YFLogxK

U2 - 10.1016/0925-7721(93)90018-2

DO - 10.1016/0925-7721(93)90018-2

M3 - Article

AN - SCOPUS:38248998602

VL - 3

SP - 277

EP - 288

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 5

ER -