
Deterministic Approximation of Random Walks in Small Space
We give a deterministic, nearly logarithmicspace algorithm that given a...
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Concentration Bounds for Cooccurrence Matrices of Markov Chains
Cooccurrence statistics for sequential data are common and important da...
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Deterministic Approximation of Random Walks via Queries in Graphs of Unbounded Size
Consider the following computational problem: given a regular digraph G=...
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Simulating Random Walks on Graphs in the Streaming Model
We study the problem of approximately simulating a tstep random walk on...
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Sparsified Block Elimination for Directed Laplacians
We show that the sparsified block elimination algorithm for solving undi...
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Spectral Sparsification of RandomWalk Matrix Polynomials
We consider a fundamental algorithmic question in spectral graph theory:...
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Transduction on Directed Graphs via Absorbing Random Walks
In this paper we consider the problem of graphbased transductive classi...
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Highprecision Estimation of Random Walks in Small Space
In this paper, we provide a deterministic Õ(log N)space algorithm for estimating the random walk probabilities on Eulerian directed graphs (and thus also undirected 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) (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 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 nonEulerian 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 randomwalk 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 "cyclelifted 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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