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.

中文翻译：

---

前往点集的中心

让磷成为一组nR中的点 d . 对于参数 α ∈ (0,1)，α 中心点为磷是一个点p∈ R d 使得所有包含的封闭半空间磷还至少包含 αn的点磷. 我们重新审视 Clarkson 等人的算法。[1996] 计算（大致）1/(4d 2)-中心点在Õ(d 9) 随机时间，其中 Õ 隐藏多对数项。我们提出了一种改进的算法，可以计算质量任意接近 1/ 的中心点d 2并以随机时间运行Õ(d 7）。虽然改进（可以说）是温和的，但这是 Clarkson 等人对算法的第一次改进。[1996] 20 多年。新算法更简单，运行时间限制遵循一个简单的随机游走参数，我们认为这是独立的兴趣。我们还介绍了改进的中心点算法的几个新应用。