2026-07-23T07:58:25Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:11257 2024-03-06T10:47:49Z ddc:004 open_access

Deterministic Approximation of Random Walks in Small Space Murtagh, Jack Reingold, Omer Sidford, Aaron Vadhan, Salil random walks space complexity derandomization spectral approximation expander graphs We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk. Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jack Murtagh and Omer Reingold and Aaron Sidford and Salil Vadhan 2019 Is Part Of LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX-RANDOM.2019.42 urn:nbn:de:0030-drops-112577 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.42 eng https://creativecommons.org/licenses/by/3.0/legalcode