Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is minimal for all points in the cell. We give novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most $(1+\varepsilon)$ times the minimum weighted distance, for any fixed parameter $\varepsilon \in (0,1)$. The diagram size is $O_d(n \log(1/\varepsilon)/\varepsilon^{d-1})$ and the construction time is within an $O_D(\log(n)/\varepsilon^{(d+5)/2})$-factor of the size bound. We also prove a matching lower bound for the size, showing that the proposed method is the first to achieve \emph{optimal size}, up to $\Theta(1)^d$-factors. In particular, the obscure $\log(1/\varepsilon)$ factor is unavoidable. As a by-product, we obtain a factor $d^{O(d)}$ improvement in size for the unweighted case and $O(d \log(n) + d^2 \log(1/\varepsilon))$ point-location time in the subdivision, improving the known query bound by one $d$-factor. The key ingredients of our approximation algorithms are the study of convex regions that we call cores, an adaptive refinement algorithm to obtain optimal size, and a novel notion of \emph{bisector coresets}, which may be of independent interest. In particular, we show that coresets with $O_d(1/\varepsilon^{(d+3)/2})$ worst-case size can be computed in near-linear time.

翻译：给定$\mathbb{R}^d$中的$n$个站点，每个站点具有正权重因子，乘法加权Voronoi图是空间的一种划分，其中每个胞元关联到使得该胞元内所有点的加权欧氏距离最小的站点。我们提出新颖的近似算法，输出基于立方体的划分，使得点与关联站点的加权距离至多为最小加权距离的$(1+\varepsilon)$倍，其中$\varepsilon \in (0,1)$为任意固定参数。该图的规模为$O_d(n \log(1/\varepsilon)/\varepsilon^{d-1})$，构造时间在$O_D(\log(n)/\varepsilon^{(d+5)/2})$因子内接近规模上界。我们还证明了规模的匹配下界，表明所提方法是首个达到\emph{最优规模}（至多相差$\Theta(1)^d$因子）的方法。特别地，$O(\log(1/\varepsilon))$因子是不可避免的。作为副产品，我们在无权重情形下获得了$d^{O(d)}$的规模改进，并在划分中实现了$O(d \log(n) + d^2 \log(1/\varepsilon))$的点定位时间，将已知查询界改进了一个$d$因子。我们近似算法的关键要素包括：对称为核的凸区域的研究、获得最优规模的自适应细化算法，以及可能具有独立意义的\emph{平分核集}新概念。特别地，我们证明了最坏情况规模为$O_d(1/\varepsilon^{(d+3)/2})$的核集可在近线性时间内计算。