Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers
Abstract
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 $d1 + O(d/\sqrt{k})$ reweighted 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 DOI:
 10.48550/arXiv.2011.08806
 arXiv:
 arXiv:2011.08806
 Bibcode:
 2020arXiv201108806J
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Optimization and Control
 EPrint:
 56 pages. Updated version includes slightly improved running time and new sparsity bounds for graph ultrasparsifiers