## Abstract

We establish a bound of O(n^{2}k^{1+ε}), for any ε > 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in ℝ^{3} with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, when k ≪ n, the new bounds on the complexity (and construction cost) of T improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set T _{ℓ0} of line transversals which emanate from a fixed line ℓ_{0}, establish an almost tight bound of O(nk^{1+ε}) on the complexity of T_{ℓ0}, and provide a randomized algorithm which computes T_{ℓ0} in comparable expected time. Slightly improved combinatorial bounds for the complexity of T_{ℓ0}, and comparable improvements in the cost, of constructing this set, are established for two special cases, both assuming that the polyhedra of P are pairwise disjoint: the case where ℓ_{0} is disjoint from the polyhedra of P, and the case where the polyhedra of P are unbounded in a direction parallel to ℓ_{0}.

Original language | English |
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Title of host publication | Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms |

Publisher | Association for Computing Machinery (ACM) |

Pages | 170-179 |

Number of pages | 10 |

ISBN (Print) | 9780898716801 |

DOIs | |

State | Published - 2009 |

Event | 20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States Duration: 4 Jan 2009 → 6 Jan 2009 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 20th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | New York, NY |

Period | 4/01/09 → 6/01/09 |

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