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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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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.

Cite As Get BibTex

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

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.

Subject Classification

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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References

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