### Abstract

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires at least degree 3. We provide an O(n log n) time algorithm for constructing such a tree, and show that this is optimal.

Original language | English (US) |
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Title of host publication | Algorithms and Computation - 5th International Symposium, ISAAC 1994, Proceedings |

Publisher | Springer-Verlag |

Pages | 20-28 |

Number of pages | 9 |

ISBN (Print) | 9783540583257 |

State | Published - Jan 1 1994 |

Event | 5th Annual International Symposium on Algorithms and Computation, ISAAC 1994 - Beijing, China Duration: Aug 25 1994 → Aug 27 1994 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 834 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 5th Annual International Symposium on Algorithms and Computation, ISAAC 1994 |
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Country | China |

City | Beijing |

Period | 8/25/94 → 8/27/94 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Computation - 5th International Symposium, ISAAC 1994, Proceedings*(pp. 20-28). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 834 LNCS). Springer-Verlag.

**Every set of disjoint line segments admits a binary tree.** / Bose, Prosenjit; Houle, Michael E.; Toussaint, Godfried.

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

*Algorithms and Computation - 5th International Symposium, ISAAC 1994, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 834 LNCS, Springer-Verlag, pp. 20-28, 5th Annual International Symposium on Algorithms and Computation, ISAAC 1994, Beijing, China, 8/25/94.

}

TY - GEN

T1 - Every set of disjoint line segments admits a binary tree

AU - Bose, Prosenjit

AU - Houle, Michael E.

AU - Toussaint, Godfried

PY - 1994/1/1

Y1 - 1994/1/1

N2 - Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires at least degree 3. We provide an O(n log n) time algorithm for constructing such a tree, and show that this is optimal.

AB - Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires at least degree 3. We provide an O(n log n) time algorithm for constructing such a tree, and show that this is optimal.

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UR - http://www.scopus.com/inward/citedby.url?scp=85027522589&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85027522589

SN - 9783540583257

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

SP - 20

EP - 28

BT - Algorithms and Computation - 5th International Symposium, ISAAC 1994, Proceedings

PB - Springer-Verlag

ER -