License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.42
URN: urn:nbn:de:0030-drops-112577
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11257/
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Murtagh, Jack ; Reingold, Omer ; Sidford, Aaron ; Vadhan, Salil

Deterministic Approximation of Random Walks in Small Space

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LIPIcs-APPROX-RANDOM-2019-42.pdf (0.6 MB)

Abstract

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.

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 =	{42:1--42:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11257},
  URN =		{urn:nbn:de:0030-drops-112577},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.42},
  annote =	{Keywords: random walks, space complexity, derandomization, spectral approximation, expander graphs}
}

Keywords: random walks, space complexity, derandomization, spectral approximation, expander graphs
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue Date: 2019
Date of publication: 17.09.2019

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