Problem 22: Random Walks

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Suggested by Rina Panigrahy
Source Kanpur 2009
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The paper of Das Sarma, Gollapudi, and Panigrahy [DasSarmaGP-08] shows a method for performing random walks in the streaming model. In particular, a random walk of length $l$ can be simulated using $O(n)$ space and $O(\sqrt{l})$ passes over the input stream. Is it possible to simulate such a random walk using $\tilde O(n)$ space and a much smaller number of passes, say, at most polylogarithmic in $n$ and $l$? The goal is either to show an algorithm or prove a lower bound.

Das Sarma et al. [DasSarmaGP-08] simulate random walks to approximate the probability distribution on the vertices of the graph after a random walk of length $l$. What is the streaming complexity of approximating this distribution? What is the streaming complexity of finding the $k$ (approximately) most likely vertices after a walk of length $l$?

Update: See some progress in [ChenKPSSY21].