The Euclidean Steiner tree problem asks to find a min-cost metric graph that connects a given set of \emph{terminal} points $X$ in $\mathbb{R}^d$, possibly using points not in $X$ which are called Steiner points. Even though near-linear time $(1 + \epsilon)$-approximation was obtained in the offline setting in seminal works of Arora and Mitchell, efficient dynamic algorithms for Steiner tree is still open. We give the first algorithm that (implicitly) maintains a $(1 + \epsilon)$-approximate solution which is accessed via a set of tree traversal queries, subject to point insertion and deletions, with amortized update and query time $O(\poly\log n)$ with high probability. Our approach is based on an Arora-style geometric dynamic programming, and our main technical contribution is to maintain the DP subproblems in the dynamic setting efficiently. We also need to augment the DP subproblems to support the tree traversal queries.

翻译：欧几里得斯坦纳树问题要求找到一条最小代价度量图，连接给定的终端点集合 $X \subseteq \mathbb{R}^d$，并允许使用不在 $X$ 中的斯坦纳点。尽管在离线场景下，Arora和Mitchell的开创性工作中已获得近线性时间 $(1 + \epsilon)$ 近似算法，但针对斯坦纳树的高效动态算法仍是一个开放问题。我们首次提出一种算法，能在点插入和删除操作下（隐式地）维护一个 $(1 + \epsilon)$ 近似解，并通过一组树遍历查询来访问该解，其均摊更新和查询时间为 $O(\poly\log n)$（高概率）。我们的方法基于Arora风格的几何动态规划，主要技术贡献在于高效维护动态场景下的动态规划子问题，同时还需扩展这些子问题以支持树遍历查询。