We give a deterministic, nearly logarithmic-space algorithm that given an undirected multigraph $G$, 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, RANDOM'05), which we recently used for solving Laplacian systems in nearly logarithmic space (Murtagh et al., FOCS'17), 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 multigraphs and for sparsifying the product of two distinct multigraphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size.

----------

A conference version of this paper appeared in the Proceedings of the 23rd International Conference on Randomization and Computation (RANDOM 2019).