2 research outputs found
Lower bounds for distributed markov chain problems
We study the worst-case communication complexity of distributed algorithms
computing a path problem based on stationary distributions of random walks in a
network with the caveat that is also the communication network. The
problem is a natural generalization of shortest path lengths to expected path
lengths, and represents a model used in many practical applications such as
pagerank and eigentrust as well as other problems involving Markov chains
defined by networks.
For the problem of computing a single stationary probability, we prove an
bits lower bound; the trivial centralized algorithm costs
bits and no known algorithm beats this. We also prove lower bounds for
the related problems of approximately computing the stationary probabilities,
computing only the ranking of the nodes, and computing the node with maximal
rank. As a corollary, we obtain lower bounds for labelling schemes for the
hitting time between two nodes
Distributed Random Walks
Performing random walks in networks is a fundamental primitive that has found
applications in many areas of computer science, including distributed
computing. In this paper, 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 as
a subroutine always do so naively, i.e., in rounds. The main
contribution of this paper 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 in
rounds ( hides \polylog{n} factors where is the number of
nodes in the network) with high probability on an undirected network, where
is the diameter of the network. For small diameter graphs, this is a
significant improvement over the naive bound. Furthermore, our
algorithm is optimal within a poly-logarithmic factor as there exists a
matching lower bound [Nanongkai et al. PODC 2011]. We further extend our
algorithms to efficiently perform independent random walks in
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, such as computing a random
spanning tree and estimating mixing time and related parameters. Our algorithm
is fully decentralized and can serve as a building block in the design of
topologically-aware networks.Comment: Preprint of an article to appear in Journal of the ACM in February
2013. The official journal version has several gramatical corrections.
Preliminary versions of this paper appeared in PODC 2009 and PODC 2010. arXiv
admin note: substantial text overlap with arXiv:0911.3195, arXiv:1205.552