APPROXIMATING CENTER POINTS WITH ITERATIVE RADON POINTS

KENNETH L. CLARKSON, DAVID EPPSTEIN, GARY L. MILLER, CARL STURTIVANT, SHANG-HUA TENG
1996 International journal of computational geometry and applications  
We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d . A point c ∈ IR d is a β-center point of P if every closed halfspace containing c contains at least βn points of P . Every point set has a 1/(d + 1)-center point; our algorithm finds an Ω(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.
more » ... over, it can be optimally parallelized to require O(log 2 d log log n) 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 -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. Introduction A center point of a set P of n points in IR d is a point c of IR d such that every hyperplane passing through c partitions P into two subsets each of size at most nd/(d + 1) [9, 27] . This balanced separation property makes the center point useful for efficient divide-and-conquer algorithms in geometric computing [1, 18, 20, 27] and large-scale scientific computing [19, 23] . Recently, Donoho and Gasko [8] have suggested center points as estimators for multivariate datasets. They showed such points as estimators are robust and have a high "breakdown point." Note that we are not * The preliminary version of this paper appeared in the 9th ACM Symp. Computational Geometry, (1993).
doi:10.1142/s021819599600023x fatcat:tp4mz4vz2bf6rf2oujhinqxnfi