Distributed Random Walks

  • Authors:

  • Atish Das Sarma;Danupon Nanongkai;Gopal Pandurangan;Prasad Tetali

  • Affiliations:

  • eBay Research Labs;Nanyang Technological University;Nanyang Technological University and Brown University;Georgia Institute of Technology

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 2013

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Abstract

Performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science, including distributed computing. In this article, we focus on the problem of sampling random walks efficiently in a distributed network and its applications. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain random walk samples. All previous algorithms that compute a random walk sample of length ℓ as a subroutine always do so naively, that is, in O(ℓ) rounds. The main contribution of this article is a fast distributed algorithm for performing random walks. We present a 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 ℓ in Õ(√ℓD) rounds (Õ hides polylog n factors where n is the number of nodes in the network) with high probability on an undirected network, where D is the diameter of the network. For small diameter graphs, this is a significant improvement over the naive O(ℓ) bound. Furthermore, our algorithm is optimal within a poly-logarithmic factor as there exists a matching lower bound [Nanongkai et al. 2011]. We further extend our algorithms to efficiently perform k independent random walks in Õ(√kℓD + k) rounds. We also show that our algorithm can be applied to speedup the more general Metropolis-Hastings sampling. Our random-walk algorithms can be used to speed up distributed algorithms in 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 unweighted) network which runs in Õ(√mD) rounds with high probability (m 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.