## 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 we 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 Tℓ_{0} 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}. Our result is related to the problem of bounding the number of geometric permutations of a collection C of k pairwise-disjoint convex sets in ℝ^{3}, namely, the number of distinct orders in which the line transversals of C visit its members. We obtain a new partial result on this problem.

Original language | English |
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Pages (from-to) | 3283-3310 |

Number of pages | 28 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 7 |

DOIs | |

State | Published - 1 Sep 2010 |

Externally published | Yes |

## Keywords

- Combinatorial complexity
- Convex polyhedra
- Extremal stabbing lines
- Line transversals
- Lines in space

## ASJC Scopus subject areas

- Computer Science (all)
- Mathematics (all)

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