# Deterministic Approximation of Random Walks in Small Space

## TL;DR

This paper presents a deterministic, space-efficient algorithm for approximating conductance and spectral properties of random walks on graphs, improving efficiency and generality over previous randomized methods.

## Contribution

It introduces a nearly logarithmic-space deterministic algorithm for approximating conductance and spectral properties of r-step random walks, generalizing derandomized graph operations.

## Key findings

- Deterministic algorithm approximates conductance within 1+ε factor.
- Provides nearly linear-time randomized spectral approximation for odd r.
- Generalizes derandomized square to irregular graphs and graph products.

## 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 (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.