We propose a self-improving algorithm for computing Voronoi diagrams under a given convex distance function with constant description complexity. The $n$ input points are drawn from a hidden mixture of product distributions; we are only given an upper bound $m = o(\sqrt{n})$ on the number of distributions in the mixture, and the property that for each distribution, an input instance is drawn from it with a probability of $\Omega(1/n)$. For any $\varepsilon \in (0,1)$, after spending $O\bigl(mn\log^{O(1)} (mn) + m^{\varepsilon} n^{1+\varepsilon}\log(mn)\bigr)$ time in a training phase, our algorithm achieves an $O\bigl(\frac{1}{\varepsilon}n\log m + \frac{1}{\varepsilon}n2^{O(\log^* n)} + \frac{1}{\varepsilon}H\bigr)$ expected running time with probability at least $1 - O(1/n)$, where $H$ is the entropy of the distribution of the Voronoi diagram output. The expectation is taken over the input distribution and the randomized decisions of the algorithm. For the Euclidean metric, the expected running time improves to $O\bigl(\frac{1}{\varepsilon}n\log m + \frac{1}{\varepsilon}H\bigr)$.

翻译：我们提议在给定的 convex 距离函数下计算 Voronoi 图表的自我改进算法 。 $n 输入点来自产品分布的隐藏混合 。 在培训阶段, 我们只得到该混合物分布数量的上限$ = o( sqrt{n} 美元, 而对于每次分布的属性, 输入实例从它中得出, 概率为$\ Omega( 1/ n) 。 对于任何在花费 $gigl( mn\ log_ O(1)} (mn) 之后的 美元输入点。 在花费 $gigl (m) = = o( varepsl> {m)\ bigr) $; 在培训阶段, 我们的算法达到 $O\ bigl( g) (l) {1\\\ varepsilon} n_O\\\\\\\\\\\ r\\\ r\\\\ r\ r\ rql=l=l=l= r= ral r=l= r=l=l=l=l=l= r=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l= r=l=l=l=l=l=l=l=l=l=l=l= r=l=l=l= rxxxxxxxl=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=n=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=l=