Let $S$ be a set of $n$ sites, each associated with a point in $\mathbb{R}^2$ and a radius $r_s$ and let $\mathcal{D}(S)$ be the disk graph on $S$. We consider the problem of designing data structures that maintain the connectivity structure of $\mathcal{D}(S)$ while allowing the insertion and deletion of sites. For unit disk graphs we describe a data structure that has $O(\log^2n)$ amortized update time and $O((\log n)/(\log\log n))$ amortized query time. For disk graphs where the ratio $\Psi$ between the largest and smallest radius is bounded, we consider the decremental and the incremental case separately, in addition to the fully dynamic case. In the fully dynamic case we achieve amortized $O(\Psi \lambda_6(\log n) \log^{9}n)$ update time and $O(\log n)$ query time, where $\lambda_s(n)$ is the maximum length of a Davenport-Schinzel sequence of order $s$ on $n$ symbols. This improves the update time of the currently best known data structure by a factor of $\Psi$ at the cost of an additional $O(\log \log n)$ factor in the query time. In the incremental case we manage to achieve a logarithmic dependency on $\Psi$ with a data structure with $O(\alpha(n))$ query and $O(\log\Psi \lambda_6(\log n) \log^{9}n)$ update time. For the decremental setting we first develop a new dynamic data structure that allows us to maintain two sets $B$ and $P$ of disks, such than at a deletion of a disk from $B$ we can efficiently report all disks in $P$ that no longer intersect any disk of $B$. Having this data structure at hand, we get decremental data structures with an amortized query time of $O((\log n)/(\log \log n))$ supporting $m$ deletions in $O((n\log^{5}n + m \log^{9}n) \lambda_6(\log n) + n\log\Psi\log^4n)$ overall time for bounded radius ratio $\Psi$ and $O(( n\log^{6} n + m \log^{10}n) \lambda_6(\log n))$ for general disk graphs.

中文翻译：

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磁盘图中的动态连接

设 $S$ 是一组 $n$ 个站点，每个站点与 $\mathbb{R}^2$ 中的一个点和半径 $r_s$ 相关联，并让 $\mathcal{D}(S)$ 是磁盘图在 $S$。我们考虑设计数据结构的问题，该数据结构保持 $\mathcal{D}(S)$ 的连接结构，同时允许站点的插入和删除。对于单位磁盘图，我们描述了一个数据结构，它具有 $O(\log^2n)$ 摊销更新时间和 $O((\log n)/(\log\log n))$ 摊销查询时间。对于最大和最小半径之间的比率 $\Psi$ 有界的磁盘图，除了完全动态的情况外，我们还分别考虑了递减和递增情况。在完全动态的情况下，我们实现了摊销的 $O(\Psi \lambda_6(\log n) \log^{9}n)$ 更新时间和 $O(\log n)$ 查询时间，其中 $\lambda_s(n)$ 是 $n$ 符号上 $s$ 阶的 Davenport-Schinzel 序列的最大长度。这将当前最知名的数据结构的更新时间缩短了 $\Psi$，但代价是查询时间增加了 $O(\log \log n)$ 因子。在增量情况下，我们设法实现对 $\Psi$ 的对数依赖，其数据结构具有 $O(\alpha(n))$ 查询和 $O(\log\Psi \lambda_6(\log n) \log^ {9}n)$ 更新时间。对于递减设置，我们首先开发了一种新的动态数据结构，它允许我们维护两组 $B$ 和 $P$ 磁盘，例如从 $B$ 中删除磁盘时，我们可以有效地报告 $P 中的所有磁盘$ 不再与 $B$ 的任何磁盘相交。手头有这个数据结构，