2026-07-22T16:58:04Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:7590 2024-03-06T10:40:41Z ddc:004 open_access

Probabilistic Logarithmic-Space Algorithms for Laplacian Solvers Doron, Dean Le Gall, François Ta-Shma, Amnon Laplacian solvers Randomized logspace Bounded-space complexity classes Random walks Matrix computation A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx=b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the problem. We further extend the algorithm to other families of graphs like Eulerian graphs (and directed regular graphs) and graphs that mix in polynomial time. Our approach is to pseudo-invert the Laplacian, by first "peeling-off" the problematic kernel of the operator, and then to approximate the inverse of the remaining part by using a Taylor series. We approximate the Taylor series using a previous work and the special structure of the problem. For directed graphs we exploit in the analysis the Jordan normal form and results from matrix functions. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dean Doron and François Le Gall and Amnon Ta-Shma 2017 Is Part Of LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX-RANDOM.2017.41 urn:nbn:de:0030-drops-75908 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.41 eng https://creativecommons.org/licenses/by/3.0/legalcode