### Abstract

We investigate the complexity of the outer face in arrangements of line segments of a fixed length ℓ in the plane, drawn uniformly at random within a square. We derive upper bounds on the expected complexity of the outer face, and establish a certain phase transition phenomenon during which the expected complexity of the outer face drops sharply as a function of the total number of segments. In particular we show that up till the phase transition the complexity of the outer face is almost linear in n, and that after the phase transition, the complexity of the outer face is roughly proportional to √n. Our study is motivated by the analysis of a practical point-location algorithm (so-called walk-along-a-line point-location algorithm) and indeed, it explains experimental observations of the behavior of the algorithm on arrangements of random segments.

Original language | English (US) |
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Title of host publication | Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08 |

Pages | 69-78 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 12 2008 |

Externally published | Yes |

Event | 24th Annual Symposium on Computational Geometry, SCG'08 - College Park, MD, United States Duration: Jun 9 2008 → Jun 11 2008 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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### Other

Other | 24th Annual Symposium on Computational Geometry, SCG'08 |
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Country | United States |

City | College Park, MD |

Period | 6/9/08 → 6/11/08 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Keywords

- Algorithms
- Theory

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## Cite this

*Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08*(pp. 69-78). (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1377676.1377689