Improved Tradeoffs for Leader Election

We consider leader election in clique networks, where n nodes are connected by point-to-point communication links. For the synchronous clique under simultaneous wake-up, i.e., where all nodes start executing the algorithm in round 1, we show a tradeoff between the number of messages and the amount of time. The previous lower bound side of such a tradeoff, in the seminal paper of Afek and Gafni (1991), was shown only assuming adversarial wake-up. Interestingly, our new tradeoff also improves the previous lower bounds for a large part of the spectrum, even under simultaneous wake-up. More specifically, we show that any deterministic algorithm with a message complexity of n f (n) requires Omega(log n/log f(n) + 1) rounds, for f(n) > 1. Our result holds even if the node IDs are chosen from a relatively small set of size Theta(n log n), as we are able to avoid using Ramsey's theorem, in contrast to many existing lower bounds for deterministic algorithms. We also give an upper bound that improves over the previously-best tradeoff achieved by the algorithm of Afek and Gafni. Our second contribution for the synchronous clique under simultaneous wake-up is to show that Omega(n log n) is in fact a lower bound on the message complexity that holds for any deterministic algorithm with a termination time T (n) (i.e., any function of n), for a sufficiently large ID space. We complement this result by giving a simple deterministic algorithm that achieves leader election in sublinear time while sending only o(n log n) messages, if the ID space is of at most linear size. We also show that Las Vegas algorithms (that never fail) require Theta(n) messages. This exhibits a gap between Las Vegas and Monte Carlo algorithms. For the synchronous clique under adversarial wake-up, we show that Omega(n(3/2)) is a lower bound for 2-round algorithms. Our result is the first superlinear lower bound for randomized leader election algorithms in the clique. We also give a simple algorithm that matches this bound. Finally, we turn our attention to the asynchronous clique: Assuming adversarial wake-up, we give a randomized algorithm that, for any k is an element of [2, O( log n/log log n)], achieves a message complexity of O(n(1+1/k)) and an asynchronous time complexity of k + 8. Our algorithm achieves the first tradeoff between messages and time in the asynchronous model. For simultaneous wake-up, we translate the deterministic tradeoff algorithm of Afek and Gafni to the asynchronous model, thus partially answering an open problem they pose.