In this paper, we look at the problem of randomized leader election in synchronous distributed networks with a special focus on the message complexity. We provide an algorithm that solves the implicit version of leader election (where non-leader nodes need not be aware of the identity of the leader) in any general network with $O(sqrt{n} log^{7/2} n cdot t_{mix})$ messages and in $O(t_{mix}log^2 n)$ time, where $n$ is the number of nodes and $t_{mix}$ refers to the mixing time of a random walk in the network graph $G$. For several classes of well-connected networks (that have a large conductance or alternatively small mixing times e.g. expanders, hypercubes, etc), the above result implies extremely efficient (sublinear running time and messages) leader election algorithms. Correspondingly, we show that any substantial improvement is not possible over our algorithm, by presenting an almost matching lower bound for randomized leader election. We show that $Omega(sqrt{n}/phi^{3/4})$ messages are needed for any leader election algorithm that succeeds with probability at least $1-o(1)$, where $phi$ refers to the conductance of a graph. To the best of our knowledge, this is the first work that shows a dependence between the time and message complexity to solve leader election and the connectivity of the graph $G$, which is often characterized by the graphs conductance $phi$. Apart from the $Omega(m)$ bound in [Kutten et al., J.ACM 2015] (where $m$ denotes the number of edges of the graph), this work also provides one of the first non-trivial lower bounds for leader election in general networks.
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