Networks cannot compute their diameter in sublinear time

Silvio Frischknecht, Stephan Holzer, Roger Wattenhofer

Abstract: We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a (3/2 − ε)-approximation of the diameter requires rounds. Furthermore we use our new technique to prove an lower bound on approximating the girth of a graph by a factor 2 − ε.