On the complexity of asynchronous gossip

In this paper, we study the complexity of gossip in an asynchronous, message-passing fault-prone distributed system. In short, we show that an adaptive adversary can significantly hamper the spreading of a rumor, while an oblivious adversary cannot. This latter fact implies that there exist message-efficient asynchronous (randomized) consensus protocols, in the context of an oblivious adversary.

In more detail, we summarize our results as follows. If the adversary is adaptive, we show that a randomized asynchronous gossip algorithm cannot terminate in fewer than O(f(d + delta)) time steps unless Omega(n+f²) messages are exchanged, where n is the total number of processes, f is the number of tolerated crash failures, d is the maximum communication delay for the specific execution in question, and delta is the bound on relative process speeds in the specific execution. The lower bound result is to be contrasted with deterministic synchronous gossip algorithms that, even against an adaptive adversary, require only O(polylog n) time steps and O(n polylog n) messages.

In the case of an oblivious adversary, we present three different randomized, asynchronous algorithms that provide different trade-offs between time complexity and message complexity. The first algorithm is based on the epidemic paradigm, and completes in O(n / (n-f) log² n (d + δ)) time steps using O(n log³ n (d + δ)) messages, with high probability. The second algorithm relies on more rapid dissemination of the rumors, yielding a constant-time (w.r.t. n) gossip protocol: for every constant epsilon < 1, and for f ≤ n/2, there is a variant with time complexity O((1 / ε)(d+δ)) and message complexity O((1/ε)n^1+εlog n (d+δ)). The third algorithm solves a weaker version of the gossip problem in which each process receives at least a majority of the rumors. This algorithm achieves constant O(d+δ) time complexity and message complexity O(n^7/4 log² n).

As an application of these message-efficient gossip protocols, we present three randomized consensus protocols. Our consensus algorithms derive from combining each of our gossip protocols with the Canetti-Rabin framework, resulting in message-efficient consensus algorithms. The resulting protocols have time and message-complexity asymptotically equal to our gossip protocols. We particularly highlight the third consensus protocol, a result that is interesting in its own right: the first asynchronous randomized consensus algorithm with strictly subquadradic message-complexity, i.e., O(n^7/4 log² n).