Sariel Har-Peled
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/(4d2)-centerpoint in Õ(d9) randomized time, where Õ hides polylogarithmic terms. We present an improved algorithm that can compute centerpoints with quality arbitrarily close to 1/d2 and runs in randomized time Õ(d7). 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.
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Index Terms
- Journey to the Center of the Point Set
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