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**An optimal algorithm for the boundary of a cell in a union of rays.**
*(English)*
Zbl 0697.68030

Summary: We study a cell of the subdivision induced by a union of n half-lines (or rays) in the plane. We present two results. The first one is a novel proof of the O(n) bound on the number of edges of the boundary of such a cell, which is essentially of methodological interest. The second is an algorithm for constructing the boundary of any cell, which runs in optimal \(\Theta\) (n log n) time. A by-product of our results are the notions of skeleton and of skeletal order, which may be of interest in their own right.

### MSC:

68Q25 | Analysis of algorithms and problem complexity |

68U99 | Computing methodologies and applications |

68R99 | Discrete mathematics in relation to computer science |

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\textit{P. Alevizos} et al., Algorithmica 5, No. 4, 573--590 (1990; Zbl 0697.68030)

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### References:

[1] | P. Alevizos, J. D. Boissonnat, and M. Yvinec: An OptimalO(n logn) Algorithm for Contour Reconstruction from Rays,Proc. 3rd ACM Symposium on Computational Geometry, Waterloo, pp. 162-170, June 1987. |

[2] | M. Atallah: Dynamic Computational Geometry,Proc. 24th IEEE Symposium on Foundations of Computer Science, pp. 92-99, Oct. 1983. |

[3] | B. Chazelle and L. Guibas: Visibility and Intersection Problems in Plane Geometry,Proc. 1st ACM Symposium on Computational Geometry, Baltimore, pp. 135-147, June 1985. · Zbl 0695.68033 |

[4] | Chazelle, B.; Guibas, L.; Lee, D. T., The Power of Geometric Duality, BIT, 25, 76-90 (1985) · Zbl 0603.68072 |

[5] | H. Edelsbrunner, L. J. Guibas, and M. Sharir: The Complexity of Many Faces in Arrangements of Lines and Segments,Proc. 4th ACM Symposium on Computational Geometry, Urbana, pp. 44-56, June 1988. · Zbl 0691.68035 |

[6] | Edelsbrunner, H.; O’Rourke, J.; Seidel, R., Constructing Arrangements of Lines and Hyper-planes with Applications, SIAM J. Comput., 15, 341-363 (1986) · Zbl 0603.68104 |

[7] | L. J. Guibas, M. Sharir, and S. Sifrony: On the General Motion Planning Problem with Two Degrees of Freedom,Proc. 4th ACM Symposium on Computational Geometry, Urbana, pp. 319-329, June 1988. · Zbl 0685.68049 |

[8] | Hart, S.; Sharir, M., Non-Linearity of Davenport-Schinzel Sequences and of Generalized Path Compression Schemes, Combinatorica, 6, 2, 151-177 (1986) · Zbl 0636.05003 |

[9] | Pollack, R.; Sharir, M.; Sifrony, S., Separating Two Simple Polygons by a Sequence of Translations, Discrete Comput. Geom., 3, 123-136 (1988) · Zbl 0646.68052 |

[10] | Wiernik, A.; Sharir, M., Planar Realizations of Nonlinear Davenport-Schinzel Sequences by Segments, Discrete Comput. Geom., 3, 15-47 (1988) · Zbl 0636.68043 |

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