Approximating center points with iterative radon points

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in R(d). A point c is an element of R(d) is a beta-center point of P if every closed halfspace containing c contains at least pn points of P. Every point set has a 1/(d + 1)-center point; our algorithm finds an Omega(1/d(2))-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log(2) dloglogn) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak epsilon-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly.