# Deterministic approximation of random walks in small space

### Citation:

Murtagh, Jack, Omer Reingold, Aaron Sidford, and Salil Vadhan. “Deterministic approximation of random walks in small space.” Theory of Computing Special Issue on RANDOM '19 (Forthcoming).
 TOC 2020.pdf 341 KB APPROX-RANDOM 2019 590 KB ArXiv2019.pdf 369 KB ArXiv 2019 v2.pdf 0 bytes

### Abstract:

Version History: v1, 15 Mar. 2019: https://arxiv.org/abs/1903.06361v1
v2 in ArXiv, 25 Nov. 2019: https://arxiv.org/abs/1903.06361v2

Prior Published Version (APPROX-RANDOM 2019), 20 Sep 2019:
In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), Dimitris Achlioptas and László A. Végh (Eds.). Vol. 145. Cambridge, Massachusetts (MIT): Leibniz International Proceedings in Informatics (LIPIcs), 2019.

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 $$1+ϵ$$, where $$ϵ > 0$$ is an arbitrarily small constant. More generally, our algorithm computes an $$ϵ$$-spectral approximation to the normalized Laplacian of the $$r$$-step walk. Our algorithm combines the derandomized square graph operation (Rozenman and Vadhan, 2005), which we recently used for solving Laplacian systems in nearly logarithmic space (Murtagh, Reingold, Sidford, and Vadhan, 2017), with ideas from (Cheng, Cheng, Liu, Peng, and Teng, 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.

Publisher's Version

Last updated on 04/06/2021