Coordinated Consensus in Dynamic Networks

We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worst-case adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and Delta-coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution. Next, we show that simultaneous consensus can never be achieved in less than n - 1 rounds in any execution, where n is the size of the network; consequently, simultaneous consensus is as hard as computing an upper bound on the number of nodes in the network. For Delta-coordinated consensus, we show that if the ratio between nodes with input 0 and input 1 is bounded away from 1, it is possible to decide in time n - circle minus(root n Delta), where Delta bounds the time from the first decision until all nodes decide. If the dynamic graph has diameter D. the time to decide is min{O(nD/Delta), n - Omega(n Delta/D)}, even if D is not known in advance. Finally, we show that (a) there is a dynamic graph such that for every input, no node can decide before time n - O(Delta(0.28)n(0.72)); and (b) for any diameter D = O(Delta), there is an execution with diameter D where no node can decide before time Omega(nD/Delta). To our knowledge, our work constitutes the first study of Delta-coordinated consensus in general graphs.