We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $L_p$-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $O(n \log^3 n)$ storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required $O(n^\varepsilon)$ time for an update and $O(\log n)$ time for a query [SICOMP, 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an $O(n^\varepsilon)$ factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph.

中文翻译：

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一般距离函数的动态平面 Voronoi 图及其算法应用

我们描述了一种新的数据结构，用于关于一般距离函数族的平面中动态最近邻查询。这些包括 $L_p$-norms 和加法加权欧几里德距离。我们的数据结构支持具有恒定描述复杂度（例如，点、线段、圆盘等）的一般（凸面、成对不相交）站点。我们的结构使用 $O(n \log^3 n)$ 存储，并且需要多对数更新和查询时间，改进了 Agarwal、Efrat 和 Sharir 的早期数据结构，需要 $O(n^\varepsilon)$ 时间进行更新和 $O(\log n)$ 查询时间 [SICOMP, 1999]。我们的数据结构有很多应用。在所有这些中，它提供了更快的算法，通常将先前边界中的 $O(n^\varepsilon)$ 因子减少到多对数。此外，我们在这里给出了两个新的应用程序：