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...

• Type: Conference
• Year: 2007
• Where: COMPGEOM
• Authors: Boris Aronov, Sariel Har-Peled, Micha Sharir
• Keywords: COMPGEOM 2007, Discrete Geometry, Range Spaces, Relative -approximations, Smaller-size Relative -approximations