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Abstract

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 + ε, where ε > 0 is an arbitrarily small constant. More generally, our algorithm computes an ε-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

Improved analysis of higher order random walks and applications

“…Since then, the derandomized squaring algorithm has also found other uses in derandomization, e.g. [MRSV17,MRSV21]. We conclude by noting that the inital algorithm of [Rei08] was based on the zigzag product construction [RVW00], which has also inspired research in the field of high dimensional expansion [KK20].…”