Murtagh, Jack ;
Reingold, Omer ;
Sidford, Aaron ;
Vadhan, Salil
Deterministic Approximation of Random Walks in Small Space
Abstract
We give a deterministic, nearly logarithmicspace algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the rstep 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 epsilonspectral approximation to the normalized Laplacian of the rstep 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 timeefficient (while ours is spaceefficient) 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 lineartime 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.
BibTeX  Entry
@InProceedings{murtagh_et_al:LIPIcs:2019:11257,
author = {Jack Murtagh and Omer Reingold and Aaron Sidford and Salil Vadhan},
title = {{Deterministic Approximation of Random Walks in Small Space}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {42:142:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771252},
ISSN = {18688969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11257},
URN = {urn:nbn:de:0030drops112577},
doi = {10.4230/LIPIcs.APPROXRANDOM.2019.42},
annote = {Keywords: random walks, space complexity, derandomization, spectral approximation, expander graphs}
}
17.09.2019
Keywords: 

random walks, space complexity, derandomization, spectral approximation, expander graphs 
Seminar: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Issue date: 

2019 
Date of publication: 

17.09.2019 