  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.

Data Structures and Algorithms;Optimization and Control