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Author Ezra, Esther ♦ Sharir, Micha
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2009
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword (1/r)-cuttings ♦ Union of simply-shaped bodies ♦ Curve-sensitive cuttings ♦ Hierarchical decomposition of convex polytopes
Abstract We show that the combinatorial complexity of the union of $\textit{n}$ “fat” tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is $O(n^{2+ϵ}),$ for any ϵ > 0;the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [2003]. Our result extends, in a significant way, the result of Pach et al. [2003] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with 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 $\textit{dihedral}$ wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of $\textit{n}$ cubes in $R^{3},$ having arbitrary side lengths, is $O(n^{2+ϵ}),$ for any ϵ > 0 (again, significantly extending the result of Pach et al. [2003]). Finally, 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 $R^{3}.$
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2009-11-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 57
Issue Number 1
Page Count 23
Starting Page 1
Ending Page 23


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Source: ACM Digital Library