Let $S \subseteq \mathbb{R}^2$ be a set of $n$ \emph{sites} in the plane, so that every site $s \in S$ has an \emph{associated radius} $r_s > 0$. Let $D(S)$ be the \emph{disk intersection graph} defined by $S$, i.e., the graph with vertex set $S$ and an edge between two distinct sites $s, t \in S$ if and only if the disks with centers $s$, $t$ and radii $r_s$, $r_t$ intersect. Our goal is to design data structures that maintain the connectivity structure of $D(S)$ as $S$ changes dynamically over time. We consider the incremental case, where new sites can be inserted into $S$. While previous work focuses on data structures whose running time depends on the ratio between the smallest and the largest site in $S$, we present a data structure with $O(\alpha(n))$ amortized query time and $O(\log^6 n)$ expected amortized insertion time.

翻译：设 $S \subseteq \mathbb{R}^2$ 为平面上的 $n$ 个*站点*集合，且每个站点 $s \in S$ 具有*关联半径* $r_s > 0$。令 $D(S)$ 为由 $S$ 定义的*圆盘相交图*，即顶点集为 $S$ 的图，其中两个不同站点 $s, t \in S$ 之间存在边当且仅当以 $s$、$t$ 为圆心、半径分别为 $r_s$、$r_t$ 的圆盘相交。我们的目标是设计数据结构，在 $S$ 随时间动态变化时维护 $D(S)$ 的连通结构。我们考虑增量情形，即新站点可以插入到 $S$ 中。现有工作依赖于 $S$ 中最小与最大站点半径之比的数据结构运行时间，而我们提出的数据结构可实现 $O(\alpha(n))$ 的均摊查询时间以及 $O(\log^6 n)$ 的期望均摊插入时间。