Almost Tight Bounds For Rumour Spreading With Conductance

We show that if a connected graph with n nodes has conductance phi then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within (O) over tilde(phi(-1) . log n), many rounds with high probability, regardless of the source, by using the PUSH-PULL strategy. The (O) over tilde(...) notation hides a polylog phi(-1) factor. This result is almost tight since there exists graph of n nodes, and conductance 0, with diameter Omega(phi(-1) . log n).If, in addition, the network satisfies some kind of uniformity condition on the degrees, our analysis implies that both both PUSH and PULL, by themselves, successfully broadcast the message to every node in the same number of rounds.