In this paper we provide an $O(m (\log \log n)^{O(1)} \log(1/\epsilon))$-expected time algorithm for solving Laplacian systems on $n$-node $m$-edge graphs, improving improving upon the previous best expected runtime of $O(m \sqrt{\log n} (\log \log n)^{O(1)} \log(1/\epsilon))$ achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of $\ell_p$-stretch graph approximations with improved stretch and sparsity bounds. Additionally, as motivation for this work, we show that for every set of vectors in $\mathbb{R}^d$ (not just those induced by graphs) and all $k > 1$ there exist ultrasparsifiers with $d-1 + O(d/\sqrt{k})$ re-weighted vectors of relative condition number at most $k$. For small $k$, this improves upon the previous best known relative condition number of $\tilde{O}(\sqrt{k \log d})$, which is only known for the graph case.

翻译：本文提出一种期望时间复杂度为 $O(m (\log \log n)^{O(1)} \log(1/\epsilon))$ 的算法，用于求解 $n$ 节点 $m$ 边图上的拉普拉斯系统。这一结果改进了此前由（Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014）实现的 $O(m \sqrt{\log n} (\log \log n)^{O(1)} \log(1/\epsilon))$ 最优期望运行时间。为获得该结果，我们构造了具有改进拉伸和稀疏性边界的 $\ell_p$-拉伸图近似。此外，作为本工作的动机，我们证明：对于 $\mathbb{R}^d$ 中的任意向量集合（不仅限于图诱导的向量）及所有 $k > 1$，存在至多包含 $d-1 + O(d/\sqrt{k})$ 个重加权向量且相对条件数不超过 $k$ 的超稀疏化。对于较小的 $k$，这一结果改进了此前仅在图情形下已知且最优相对条件数为 $\tilde{O}(\sqrt{k \log d})$ 的结论。