Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

In this paper we provide an $O(m (\log \log n)^{O(1)} \log(1/ε))$-expected time algorithm for solving Laplacian systems on $n$-node $m$-edge graphs, improving improving upon the previous best expected runtime of $O(m \sqrt{\log n} (\log \log n)^{O(1)} \log(1/ε))$ 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.