Journey to the Center of the Point Set

Abstract

Let P be a set of n points in R d . For a parameter α ∈ (0,1), an α-centerpoint of P is a point p ∈ R d such that all closed halfspaces containing P also contain at least α n points of P . We revisit an algorithm of Clarkson et al. [1996] that computes (roughly) a 1/(4 d 2 )-centerpoint in Õ( d 9 ) randomized time, where Õ hides polylogarithmic terms. We present an improved algorithm that can compute centerpoints with quality arbitrarily close to 1/ d 2 and runs in randomized time Õ( d 7 ). While the improvements are (arguably) mild, it is the first refinement of the algorithm by Clarkson et al. [1996] in over 20 years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.