Leader Election in Unreliable Radio Networks.

The dual graph model describes a radio network that contains both reliable and unreliable links. In recent years, this model has received significant attention by the distributed algorithms community [Kuhn/Lynch/Newport/Oshman/Richa, PODC 2010; Censor-Hillel/Gilbert/Kuhn/Lynch/Newport, Dist. Comp. 2014; Ghaffari/Haeupler/Lynch/Newport, DISC 2012; Ghaffari/Lynch/Newport, PODC 2013; Ghaffir/Kantor/Lynch/Newport, PODC 2014; Newport, DISC 2014; Ahmadi/Ghodselahi/Kuhn/Molla, OPODIS 2015; Lynch/Newport, PODC 2015]. Due to results in [Ghaffari/Lynch/Newport, PODC 2013], it is known that leader election plays a key role in enabling efficient computation in this difficult setting: a leader can synchronize the network in such a manner that most problems can be subsequently solved in time similar to the classical radio network model that lacks unreliable links. The feasibility of efficient leader election in the dual graph model, however, was left as an important open question. In this paper, we answer this question. In more detail, we prove new upper and lower bound results that characterize the complexity of leader election in this setting. By doing so, we reveal a surprising dichotomy: (1) under the assumption that the network size n is in the range 1 to N, where N is a large upper bound on the maximum possible network size (e.g., the ID space), leader election is fundamentally hard, requiring ~Omega(sqrt(N)) rounds to solve in the worst-case; (2) under the assumption that n is in the range 2 to N, however, the problem can be solved in only ~O(D) rounds, for network diameter D, matching the lower bound for leader election in the standard radio network model (within log factors) [Ghaffari/Haeupler, SODA 2013].