Almost tight bound for the union of fat tetrahedra in three dimensions

Esther Ezra*, Micha Sharir

*Corresponding author for this work

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

Abstract

We show that the combinatorial complexity of the union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O(n2+ε), for any ε > 0; the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combinedwith the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in IR3, having arbitrary side lengths, is O(n2+ε), for any ε ≥ 0 (again, significantly extending the result of [24]). Our analysis can easily be extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in IR3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.

Original languageEnglish
Title of host publicationProceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007
Pages525-535
Number of pages11
DOIs
StatePublished - 2007
Event48th Annual Symposium on Foundations of Computer Science, FOCS 2007 - Providence, RI, United States
Duration: 20 Oct 200723 Oct 2007

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference48th Annual Symposium on Foundations of Computer Science, FOCS 2007
Country/TerritoryUnited States
CityProvidence, RI
Period20/10/0723/10/07

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