Approximations and optimal geometric divide-and-conquer

We give an efficient deterministic algorithm for computing ?-approximations and ?-nets for range spaces of bounded VC-dimension. We assume that an n-point range space ? = (X, R) of VC-dimension d is given to us by an oracle, which given a subset A ? X, returns a list of all distinct sets of the form A ? R; R ? R (in time O(|A|d+1)). Given a parameter r, the algorithm computes a (1/r)-approximation of size O(r2 log r) for ?, in time O(n(r2 log r)d). A (1/r)-net of size O(r log r) can be computed within the same time bound. We also obtain a new deterministic algorithm which for a given collection H of n hyperplanes in Ed and a parameter r ? n computes a (1/r)-cutting of (asymptotically optimal) size O(rd). For r ? n1??, where ? > 0 is arbitrary but fixed, the running time is O(nrd?1), which is optimal for geometric divide-and-conquer applications.