Deterministic Approximation of Random Walks in Small Space

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
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 + , where > 0 is an arbitrarily small constant. More generally, our algorithm computes an -spectral approximation to the normalized Laplacian of the r-step walk. Our algorithm combines the derandomized square graph operation [21] , which we recently used for solving
more » ... lacian systems in nearly logarithmic space [16] , with ideas from [5] , 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.
doi:10.4230/lipics.approx-random.2019.42 dblp:conf/approx/MurtaghRSV19 fatcat:qbobogy745ds7b43u23fx3oj4y