Epsilon-Nets and Computational Geometry

Abstract

In this chapter, we present some results from the theory of range spaces of finite VC-dimension. We introduce canonical geometric range spaces and we indicate how other range spaces encountered in computational geometry can be embedded into the canonical ones. As applications of e-nets for computational geometry problems, we mention the Cutting lemma, the Short edge lemma and the construction of an arrangement with small zones. We then indicate some improvements of these results when the application of general e-net results is replaced by other methods, and we also address the issue of de-randomizing the algorithms based on e-nets and related probabilistic techniques. We give main ideas of several proofs.