Improving the deterministic space emulation of RL is among
the most central open problems in the area of derandomization.
The best known emulation, which uses *log to the power 3/2 space*,
dates to the mid-1990s (Saks and Zhou), and Reingold's celebrated
log-space algorithm for undirected connectivity only addresses
the most celebrated problem in RL (although it was extended to
regular directed graphs by Reingold, Trevisan and Vadhan 2006).

The current paper addresses a natural problem that generalizes
undirected connectivity and yet can be easily solved in RL:
*Given a graph, a pair of vertices (s,t), and an integer k,
approximate the probability that a k-step random walk starting
at s reaches t*.
The main result of the paper is that this problerm is almost in L
(i.e., it has a tildeO(log)-space algorithm).
The result is extended to Eulerian directed graphs
(which conceptually meets the RTV06 result).

The work builds on space-bounded Laplacian solvers, and specifically the works of Murtagh, Reingold, Sidford, and Vadhan (FOCS17 and RANDOM19). I regret not paying enough attention to these works at the time, but note that the current work addresses a computational problem that I find much more appealing, natural, and fundamental by itself.

In this paper, we provide a deterministic tildeO(logN)-space algorithm for estimating the random walk probabilities on Eulerian directed graphs (and thus also undirected graphs) to within inverse polynomial additive error (eps=1/poly(N)), where N is the length of the input. Previously, this problem was known to be solvable by a randomized algorithm using space O(logN) (Aleliunas et al., FOCS `79) and by a deterministic algorithm using space O(log^{3/2}N) (Saks and Zhou, FOCS95 and JCSS99), both of which held for arbitrary directed graphs but had not been improved even for undirected graphs.

We also give improvements on the space complexity of both of these previous algorithms for non-Eulerian directed graphs when the error (esp) is negligible, generalizing what Hoza and Zuckerman (FOCS18) recently showed for the special case of distinguishing whether a random walk probability is 0 or greater than eps.

We achieve these results by giving new reductions between powering Eulerian random-walk matrices and inverting Eulerian Laplacian matrices, providing a new notion of spectral approximation for Eulerian graphs that is preserved under powering, and giving the first deterministic tildeO(logN)-space algorithm for inverting Eulerian Laplacian matrices. The latter algorithm builds on the work of Murtagh et al. (FOCS17) that gave a deterministic tildeO(logN)-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al. (FOCS19) that gave a randomized tildeO(N)-time algorithm for inverting Eulerian Laplacian matrices. A running theme throughout these contributions is an analysis of "cycle-lifted graphs," where we take a graph and "lift" it to a new graph whose adjacency matrix is the tensor product of the original adjacency matrix and a directed cycle (or variants of one).

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