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COMPGEOM
2007
ACM

On approximate halfspace range counting and relative epsilon-approximations

9 years 3 months ago
On approximate halfspace range counting and relative epsilon-approximations
The paper consists of two major parts. In the first part, we re-examine relative -approximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. We give a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative -approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure--spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace range-counting problem in Rd with relative error , and show that relative -approximations, combined with the shallow partitioning data structures of Matousek, yields efficient solutions to this problem. For example, one of our data str...
Boris Aronov, Sariel Har-Peled, Micha Sharir
Added 14 Aug 2010
Updated 14 Aug 2010
Type Conference
Year 2007
Where COMPGEOM
Authors Boris Aronov, Sariel Har-Peled, Micha Sharir
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