**Author:** Atish Das Sarma, Danupon Nanongkai, Gopal Pandurangan, Prasad Tetali

**Download: **arXiv of the journal version, arXiv of the conference version

**Conference:** PODC 2010

**Journal:** Journal of the ACM 2013

**Abstract:**

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. We first present a fast sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk. Our algorithm performs a random walk of length in rounds (with high probability) on an undirected network, where is the diameter of the network. This improves over the previous best algorithm that ran in rounds (Das Sarma et al., PODC 2009). We further extend our algorithms to efficiently perform independent random walks in rounds. We then show that there is a fundamental difficulty in improving the dependence on any further by proving a lower bound of under a general model of distributed random walk algorithms. Our random walk algorithms are useful in speeding up distributed algorithms for a variety of applications that use random walks as a subroutine. We present two main applications. First, we give a fast distributed algorithm for computing a random spanning tree (RST) in an arbitrary (undirected) network which runs in rounds (with high probability; here is the number of edges). Our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network. Our algorithm is fully decentralized and can serve as a building block in the design of topologically-aware networks.

**Update History**

Mar 03, 2009 (New version posted on ArXiv)

Nov 06, 2009 (Link to arXiv posted)

Feb 18, 2010 (New version posted)

Feb 18, 2013 (Journal version posted)