Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces

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 $d$.