Information Spreading in Dynamic Networks Under Oblivious Adversaries

We study the problem of all-to-all information exchange, also known as 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. Gossip can be completed in linear time in any static network, but an important and basic open question for dynamic networks is the existence of a distributed protocol that can do significantly better than an easily achievable bound of O(n) rounds. In previous work, it has been shown that under adaptive adversaries —those that have full knowledge and control of the topology in every round and also have knowledge of the distributed protocol including its random choices—every token forwarding algorithm requires Ω(n/ log n) rounds to complete. In this paper, we study oblivious adversaries, which differ from adaptive adversaries in one crucial aspect— they 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. Previous work has shown that starting from a distribution in which each node has a random constant fraction of the tokens, RAND-DIFF completes in Õ(n) rounds. In contrast, we show that a polynomial slowdown is inevitable under more general distributions: we present an Ω̃(n) lower bound for RAND-DIFF under an oblivious adversary. We also present an Ω̃(n) 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) rounds with high probability, under any oblivious adversary. We also show an Õ(n) upper bound for RAND-DIFF in a restricted class of oblivious adversaries, which we call paths-respecting, that may be of independent interest.