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Deterministic Approximation of Random Walks in Small Space

Authors Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Random walks and Markov chains
Keywords
  • random walks
  • space complexity
  • derandomization
  • spectral approximation
  • expander graphs

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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.

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Jack Murtagh, Omer Reingold, Aaron Sidford, and Salil Vadhan. Deterministic Approximation of Random Walks in Small Space. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 42:1-42:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.42

Author Details

Jack Murtagh
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Omer Reingold
  • Computer Science Department, Stanford University, Stanford, CA USA
Aaron Sidford
  • Management Science & Engineering, Stanford University, Stanford, CA USA
Salil Vadhan
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Funding

  • Murtagh, Jack: Supported by NSF grant CCF-1763299.
  • Reingold, Omer: Supported by NSF grant CCF-1763311.
  • Vadhan, Salil: Supported by NSF grant CCF-1763299.

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