Approximations and Optimal Geometric Divide-and-Conquer

Abstract

We give an efficient deterministic algorithm for computing ϵ-approximations and ϵ-nets for range spaces of bounded VC-dimension. We assume that an n-point range space Σ = (X, $\text{R}$) 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 ∈ $\text{R}$ (in time O(|A|^d+1)). Given a parameter r, the algorithm computes a (1/r)-approximation of size O(r² log r) for Σ, 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 E^d and a parameter r ≤ n computes a (1/r)-cutting of (asymptotically optimal) size O(r^d). For r ≤ n^1−δ, where δ > 0 is arbitrary but fixed, the running time is O(nr^d−1), which is optimal for geometric divide-and-conquer applications.