Fast gossip via non-reversible random walk

Abstract — Distributed computation of average is essential for many tasks such as estimation, eigenvalue computation, scheduling in the context of wireless sensor and ad-hoc networks. The wireless communication imposes the gossip constraint: each node can communicate with at most one other node at a given time. Recent interest in emerging wireless sensor network has led to exciting developments in the context of gossip algorithms for averaging. Most of the known algorithms are iterative and based on certain reversible random walk on the network graph. Subsequently, the running time of algorithm is affected by the diffusive nature of reversible random walk. For example, they take Ω(n 2) time to compute average on a simple path or ring graph of n nodes. In contrast, an optimal (simple) centralized algorithm takes Θ(n) time to compute average in a path. This raises the following questions: is it possible for a distributed algorithm to compute average in O(n) time for path graph? is it possible to improve over diffusive behavior of current algorithms in arbitrary graphs? In this paper, we answer the above questions in affirmative. To overcome the diffusive nature of algorithms, we utilize non-reversible random walks. Specifically, we design our algorithms by ”projecting down ” the ”lifted ” nonreversible random walks of Diaconis-Holmes-Neal (2000) and Chen-Lovasz-Pak (1999). The running time of our algorithm is square-root of the time taken by corresponding reversible random walk for a large class of graphs including path. As a sub-routine, our algorithm uses a simple distributed maximal matching algorithm that runs in O(log 2 n) time for arbitrary graph, which may be of separate interest. I