Near-Optimal Sublinear Time Bounds for Distributed Random Walks

We focus on the problem of performing random walks efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain a random walk sample on an undirected network. Despite the widespread use of random walks in distributed computing, most algorithms that compute a random walk sample of length $\ell$ naively, i.e., in $O(\ell)$ rounds. Recently, the first sublinear time distributed algorithm was presented that ran in $\tilde{O}(\ell^{2/3}D^{1/3})$ rounds {$\tilde{O}$ hides polylog factors in the number of nodes in the network} where $D$ is the diameter of the network [Das Sarma et al. PODC 2009]. This work further conjectured that a running time of $\tilde{O}(\sqrt{\ell D})$ is possible and that this is essentially optimal. In this paper, we resolve these conjectures by showing almost tight bounds on distributed random walks. We present a distributed algorithm that performs a random walk of length $\ell$ in $\tilde{O}(\sqrt{\ell D})$ rounds, where $D$ is the diameter of the network. We then show a lower bound of $\Omega(\sqrt{\frac{\ell}{\log \ell}} + D)$ rounds for performing a walk of length $\ell$. This shows that our algorithm is near-optimal. We further extend our algorithms to perform $k$ independent random walks in roughly $\tilde{O}(\sqrt{k\ell D} + k)$ rounds. Our techniques can be useful in a variety of applications. We illustrate one such application involving the decentralized computation of mixing time of the underlying network. Our algorithms are fully decentralized and can serve as building blocks in the design of topologically-aware networks.