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. Despite the widespread use of random walks in distributed computing theory and practice for long, most algorithms that compute a random walk sample of length ℓ always do so naively, i.e., in O(ℓ) rounds. Recently, a significantly faster sublinear time distributed algorithm was presented that ran in Õ(ℓ 2/3 D 1/3) rounds 1 where D is the diameter of the network [6]. This was the first result to improve beyond linear time (in ℓ) despite the sequential nature of random walks. This work further conjectured that a running time of Õ( √ ℓD) is possible and that this is essentially optimal. In this paper, we resolve these conjectures and show almost tight bounds on the time complexity of distributed random walks. We present a fast distributed algorithm for performing random walks. Our algorithm performs a random walk of length ℓ in Õ( √ ℓD) rounds on an undirected network, where D is the diameter of the network. We then show that there is a fundamental difficulty in improving the dependence on ℓ any further by proving a lower bound of Ω( log