High-precision Estimation of Random Walks in Small Space [article]

AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, Salil Vadhan
2022 arXiv   pre-print
We provide a deterministic Õ(log N)-space algorithm for estimating random walk probabilities on undirected graphs, and more generally Eulerian directed graphs, to within inverse polynomial additive error (ϵ=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(log N) (following Aleliunas et al., FOCS 79) and by a deterministic algorithm using space O(log^3/2 N) (Saks and Zhou, FOCS 95 and JCSS 99), both of which
more » ... eld 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 is negligible (ϵ=1/N^ω(1)), generalizing what Hoza and Zuckerman (FOCS 18) recently showed for the special case of distinguishing whether a random walk probability is 0 or greater than ϵ. 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 Õ(log N)-space algorithm for inverting Eulerian Laplacian matrices. The latter algorithm builds on the work of Murtagh et al. (FOCS 17) that gave a deterministic Õ(log N)-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al. (FOCS 19) that gave a randomized Õ(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).
arXiv:1912.04524v3 fatcat:peiozxcjrff75c53cvzlx2daju