Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

TL;DR

This paper introduces a faster algorithm for solving Laplacian systems using ultrasparse ultrasparsifiers, significantly improving runtime and providing new bounds for vector sets in high-dimensional spaces.

Contribution

It presents an $O(m ( ext{log log } n)^{O(1)} ext{log}(1/ ext{epsilon}))$-time algorithm for Laplacian systems and introduces ultrasparsifiers with improved condition number bounds for vector sets.

Findings

Faster Laplacian system solver with improved expected runtime

Efficient constructions of $ ext{ell}_p$-stretch graph approximations

Existence of ultrasparsifiers with better condition number bounds for vector sets

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 $d-1 + O(d/\sqrt{k})$ 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.