Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs
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For a directed graph G with n vertices and a start vertex u_ start, we wish to (approximately) sample an L-step random walk over G starting from u_ start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be Θ̃(n · L). Prior to our work, a better space complexity was only known with Õ(√(L)) passes. We settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is Θ̃(n ·√(L)), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses Ω̃(n · L^1/p) space. In addition, we show a similar Θ̃(n ·√(L)) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph.
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