Information spreading in dynamic networks under oblivious adversaries

We study the problem of gossip in dynamic networks controlled by an adversary that can modify the network arbitrarily from one round to another, provided that the network is always connected. In the gossip problem, there are n tokens arbitrarily distributed among the n network nodes, and the goal is to disseminate all the n tokens to every node. Our focus is on token-forwarding algorithms, which do not manipulate tokens in any way other than storing, copying, and forwarding them. An important open question is whether gossip can be realized by a distributed protocol that can do significantly better than an easily achievable bound of O(n²) rounds. In this paper, we study oblivious adversaries, i.e., those that are oblivious to the random choices made by the protocol. We consider Rand-Diff, a natural distributed algorithm in which neighbors exchange a token chosen uniformly at random from the difference of their token sets. We present an Ω(n^3/2) lower bound for Rand-Diff under an oblivious adversary. We also present an Ω (n^4/3) lower bound under a stronger notion of oblivious adversary for a class of randomized distributed algorithms—symmetric knowledge-based algorithms— in which nodes make token transmission decisions based entirely on the sets of tokens they possess over time. On the positive side, we present a centralized algorithm that completes gossip in Õ(n^3/2) rounds with high probability, under any oblivious adversary. We also show an Õ (n^5/3) upper bound for Rand-Diff in a restricted class of oblivious adversaries, which we call paths-respecting, that may be of independent interest.