### Abstract

Given a set L of n disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and the tree has no crossing edges. Such a tree is known as an encompassing tree and can be constructed in O(n log n) time when no three endpoints in L are collinear. In the presence of collinear endpoints, we show first that an encompassing tree with no crossing edges exists and can be computed in O(n^{2}) time, and second that the maximum degree of a node in the minimum weight spanning tree formed by these line segments is seven, and that there exists a set of line segments achieving this bound. Finally, we show that the complexity of finding the minimum weight spanning tree is optimal Θ(n log n) when we assume that the endpoints of the line segments are in general position.

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

Pages (from-to) | 86-103 |

Number of pages | 18 |

Journal | Journal of Algorithms |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1995 |

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### ASJC Scopus subject areas

- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Journal of Algorithms*,

*19*(1), 86-103. https://doi.org/10.1006/jagm.1995.1028

**Growing a tree from its branches.** / Bose, Prosfenjit; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Journal of Algorithms*, vol. 19, no. 1, pp. 86-103. https://doi.org/10.1006/jagm.1995.1028

}

TY - JOUR

T1 - Growing a tree from its branches

AU - Bose, Prosfenjit

AU - Toussaint, Godfried

PY - 1995/1/1

Y1 - 1995/1/1

N2 - Given a set L of n disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and the tree has no crossing edges. Such a tree is known as an encompassing tree and can be constructed in O(n log n) time when no three endpoints in L are collinear. In the presence of collinear endpoints, we show first that an encompassing tree with no crossing edges exists and can be computed in O(n2) time, and second that the maximum degree of a node in the minimum weight spanning tree formed by these line segments is seven, and that there exists a set of line segments achieving this bound. Finally, we show that the complexity of finding the minimum weight spanning tree is optimal Θ(n log n) when we assume that the endpoints of the line segments are in general position.

AB - Given a set L of n disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and the tree has no crossing edges. Such a tree is known as an encompassing tree and can be constructed in O(n log n) time when no three endpoints in L are collinear. In the presence of collinear endpoints, we show first that an encompassing tree with no crossing edges exists and can be computed in O(n2) time, and second that the maximum degree of a node in the minimum weight spanning tree formed by these line segments is seven, and that there exists a set of line segments achieving this bound. Finally, we show that the complexity of finding the minimum weight spanning tree is optimal Θ(n log n) when we assume that the endpoints of the line segments are in general position.

UR - http://www.scopus.com/inward/record.url?scp=0012140328&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012140328&partnerID=8YFLogxK

U2 - 10.1006/jagm.1995.1028

DO - 10.1006/jagm.1995.1028

M3 - Article

AN - SCOPUS:0012140328

VL - 19

SP - 86

EP - 103

JO - Journal of Algorithms

JF - Journal of Algorithms

SN - 0196-6774

IS - 1

ER -