The join problem in dynamic network algorithms

Distributed algorithms in dynamic networks often employ communication patterns whose purpose is to disseminate information among the participants. Gossiping is one form of such communication pattern. In dynamic settings the set of participants can change substantially as new participants join, and as failures and voluntary departures remove those who have joined previously. A natural question for such settings is: how soon can newly joined nodes discover each other by means of gossiping? This paper abstracts and studies the Join Problem for dynamic systems that use all-to-all gossip. The problem is studied in terms of join-connectivity graphs where vertices represent the participants and where each edge represents one participant's knowledge about another. Ideally, such a graph has diameter one, i.e., all participants know each other. The diameter can grow as new participants join, and as failures remove edges from the graph. Gossip helps participants discover one another, decreasing the diameter. The results describe the lower and upper bounds on the number of communication rounds such that the participants who have previously joined discover one another, under a variety of assumptions about the joining and failures. For example, in the case when new participants join at multiple participants and participants may crash, the number of rounds cannot be bounded. In the more benign cases when the failures can be controlled or when new participants join at only one participant, the bound on rounds is shown to be logarithmic in the diameter of the initial configuration.