Simulating Random Walks on Graphs in the Streaming Model

We study the problem of approximately simulating a $t$-step random walk on a graph where the input edges come from a single-pass stream. The straightforward algorithm using reservoir sampling needs $O(nt)$ words of memory. We show that this space complexity is near-optimal for directed graphs. For undirected graphs, we prove an $Ω(n\sqrt{t})$-bit space lower bound, and give a near-optimal algorithm using $O(n\sqrt{t})$ words of space with $2^{-Ω(\sqrt{t})}$ simulation error (defined as the $\ell_1$-distance between the output distribution of the simulation algorithm and the distribution of perfect random walks). We also discuss extending the algorithms to the turnstile model, where both insertion and deletion of edges can appear in the input stream.