Relative ε-Approximations in Geometry ∗

We re-examine relative ε-approximations, previously studied in [Pol86, Hau92, LLS01], [CKMS06], and their relation to certain geometric problems. 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 ε-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. We also consider applications of the new structures for approximate range counting and related problems. ∗Work on this paper by Sariel Har-Peled was partially supported by an NSF CAREER award CCR-0132901. Work by Micha Sharir was supported by a grant from the U.S.-Israel Binational Science Foundation, by NSF Grant CCF-05-14079, by Grant 155/05 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. †Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA; sariel@uiuc.edu. ‡School of Computer Science, Tel Aviv University, Tel Aviv 69978 Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA; michas@post.tau.ac.il.

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