Line transversals of convex polyhedra in ℝ3

Haim Kaplan, Natan Rubin, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

We establish a bound of O(n2k1+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 ℝ3 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 ℓ0, establish an almost tight bound of O(nk1+ε) on the complexity of Tℓ0, and provide a randomized algorithm which computes Tℓ0 in comparable expected time. Slightly improved combinatorial bounds for the complexity of Tℓ0, 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 ℓ0 is disjoint from the polyhedra of P, and the case where the polyhedra of P are unbounded in a direction parallel to ℓ0.

Original languageEnglish
Title of host publicationProceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery (ACM)
Pages170-179
Number of pages10
ISBN (Print)9780898716801
DOIs
StatePublished - 1 Jan 2009
Externally publishedYes
Event20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States
Duration: 4 Jan 20096 Jan 2009

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference20th Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CityNew York, NY
Period4/01/096/01/09

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