  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.
