Distributed algorithms for network diameter and girth

This paper considers the problem of computing the diameter D and the girth g of an n-node network in the CONGEST distributed model. In this model, in each synchronous round, each vertex can transmit a different short (say, O(logn) bits) message to each of its neighbors. We present a distributed algorithm that computes the diameter of the network in O(n) rounds. We also present two distributed approximation algorithms. The first computes a 2/3 multiplicative approximation of the diameter in $O(D\sqrt n \log n)$ rounds. The second computes a 2−1/g multiplicative approximation of the girth in $O(D+\sqrt{gn}\log n)$ rounds. Recently, Frischknecht, Holzer and Wattenhofer [11] considered these problems in the CONGEST model but from the perspective of lower bounds. They showed an $\tilde{\Omega}(n)$ rounds lower bound for exact diameter computation. For diameter approximation, they showed a lower bound of $\tilde{\Omega}(\sqrt n)$ rounds for getting a multiplicative approximation of. Both lower bounds hold for networks with constant diameter. For girth approximation, they showed a lower bound of $\tilde{\Omega}(\sqrt n)$ rounds for getting a multiplicative approximation of on a network with constant girth. Our exact algorithm for computing the diameter matches their lower bound. Our diameter and girth approximation algorithms almost match their lower bounds for constant diameter and for constant girth.