A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, δ)$ is a graph $G$ with vertex set $P$ such that, for any pair of points $u,v \in X$, the distance between $u$ and $v$ in $G$ is at most $t$ times $δ(u,v)$. We study the problem of maintaining a spanner for a dynamic point set $X$ -- that is, when $X$ undergoes a sequence of insertions and deletions -- in a metric space of constant doubling dimension. For any constant $\varepsilon>0$, we maintain a $(1+\varepsilon)$-spanner of $P$ whose total weight remains within a constant factor of the weight of the minimum spanning tree of $X$. Each update (insertion or deletion) can be performed in $\operatorname{poly}(\log Φ)$ time, where $Φ$ denotes the aspect ratio of $X$. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

翻译：设度量空间 $(\mathcal{X}, δ)$ 中点集 $X$ 的 $t$-生成树是一个顶点集为 $P$ 的图 $G$，使得对任意两点 $u,v \in X$，$G$ 中 $u$ 与 $v$ 的距离至多为 $t$ 倍的 $δ(u,v)$。我们研究在常倍维度量空间中维护动态点集 $X$（即 $X$ 经历一系列插入和删除操作）的生成树问题。对于任意常数 $\varepsilon>0$，我们维护 $P$ 的一个 $(1+\varepsilon)$-生成树，其总权重保持在 $X$ 的最小生成树权重的常数因子内。每次更新（插入或删除）可在 $\operatorname{poly}(\log Φ)$ 时间内完成，其中 $Φ$ 表示 $X$ 的长宽比。在我们的工作之前，即使在低维欧氏空间中的点集，也未知用于维护轻量级生成树的高效动态算法。