Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

TL;DR

This paper presents a deterministic algorithm that solves undirected Laplacian linear systems in nearly logarithmic space, improving upon previous randomized and deterministic methods, and enabling efficient approximation of graph metrics.

Contribution

It introduces a nearly log-space deterministic algorithm for Laplacian systems, extending derandomization techniques beyond connectivity problems.

Findings

Deterministic $ ilde{O}( ext{log } n)$-space algorithm for Laplacian systems

Approximate solutions for hitting times, commute times, and escape probabilities

Improves space complexity from $O( ext{log}^{3/2} n)$ to nearly log space

Abstract

We give a deterministic $\tilde{O}(\log n)$-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using $O(\log n)$ space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using $O(\log^{3/2} n)$ space (Saks and Zhou, FOCS 1995 and JCSS 1999). Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC `04; Peng and Spielman, STOC `14) with ideas used to show that Undirected S-T Connectivity is in deterministic logspace (Reingold, STOC `05 and JACM `08; Rozenman and Vadhan, RANDOM `05).