Helly-type theorems for approximate covering

Julien Demouth 1 Olivier Devillers 2 Marc Glisse 1 Xavier Goaoc 1
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
2 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Let F be a covering of a unit ball U in Rd by unit balls. We prove that for any epsilon >0, the smallest subset of F leaving at most a volume epsilon of U uncovered has size O(epsilon^((1-d)/2)polylog 1/epsilon). We give an example showing that this bound is tight in the worst-case, up to a logarithmic factor, and deduce an algorithm to compute such a small subset of F. We then extend these results in several directions, including covering boxes by boxes and visibility among disjoint unit balls in R3.
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https://hal.inria.fr/inria-00179277
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Submitted on : Monday, November 5, 2007 - 10:16:07 AM
Last modification on : Friday, September 20, 2019 - 4:56:42 PM
Long-term archiving on: Friday, November 25, 2016 - 5:49:45 PM

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Julien Demouth, Olivier Devillers, Marc Glisse, Xavier Goaoc. Helly-type theorems for approximate covering. [Research Report] RR-6342, 2007, pp.12. ⟨inria-00179277v2⟩

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