  We give an algorithm that computes a $(1+\epsilon)$-approximate Steiner
forest in near-linear time $n \cdot 2^{(1/\epsilon)^{O(ddim^2)} (\log \log
n)^2}$. This is a dramatic improvement upon the best previous result due to
Chan et al., who gave a runtime of $n^{2^{O(ddim)}} \cdot
2^{(ddim/\epsilon)^{O(ddim)} \sqrt{\log n}}$.
  For Steiner tree our methods achieve an even better runtime $n (\log
n)^{(1/\epsilon)^{O(ddim^2)}}$ in doubling spaces. For Euclidean space the
runtime can be reduced to $2^{(1/\epsilon)^{O(d^2)}} n \log n$, improving upon
the result of Arora in fixed dimension $d$.
