Time and communication complexity of leader election in anonymous networks

We study the problem of randomized Leader Election in synchronous distributed networks with indistinguishable nodes. We consider algorithms that work on networks of arbitrary topology in two settings, depending on whether the size of the network, i.e., the number of nodes $n$, is known or not. In the former setting, we present a new Leader Election protocol that improves over previous work by lowering message complexity and making it close to a lower bound by a factor in $\widetilde{O}(\sqrt{t_{mix}\sqrt{\Phi}})$, where $\Phi$ is the conductance and $t_{mix}$ is the mixing time of the network graph. We then show that lacking the network size no Leader Election algorithm can guarantee that the election is final with constant probability, even with unbounded communication. Hence, we further classify the problem as Leader Election (the classic one, requiring knowledge of $n$ - as is our first protocol) or Revocable Leader Election, and present a new polynomial time and message complexity Revocable Leader Election algorithm in the setting without knowledge of network size. We analyze time and message complexity of our protocols in the CONGEST model of communication.