APPROXIMATIONS AND OPTIMAL GEOMETRIC DIVIDE-AND-CONQUER

We give an efficient deterministic algorithm for computing epsilon-approximations and epsilon-nets for range spaces of bounded VC-dimension. We assume that an n-point range space Sigma = (X,R) of VC-dimension d is given to us by an oracle, which given a subset A subset of or equal to X, returns a list of all distinct sets of the form A boolean AND R; R is an element of R (in time O(\A\(d+1))). Given a parameter r, the algorithm computes a (1/r)-approximation of size O(r(2) log r) for Sigma, in time O(n(r(2) 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 less than or equal to n computes a (1/r)-cutting of (asymptotically optimal) size O(r(d)). For r less than or equal to n(1-delta), where delta > 0 is arbitrary but fixed, the running time is O(nr(d-1)), which is optimal for geometric divide-and-conquer applications. (C) 1995 Academic Press, Inc.