Improved Tradeoffs for Leader Election
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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. More specifically, we show that any deterministic algorithm with a message complexity of n f(n) requires Ω(log n/log f(n)+1) rounds, for f(n) = Ω(log n). Our result holds even if the node IDs are chosen from a relatively small set of size Θ(nlog n), as we are able to avoid using Ramsey's theorem. We also give an upper bound that improves over the previously-best tradeoff. Our second contribution for the synchronous clique under simultaneous wake-up is to show that Ω(nlog n) is in fact a lower bound on the message complexity that holds for any deterministic algorithm with a termination time T(n). We complement this result by giving a simple deterministic algorithm that achieves leader election in sublinear time while sending only o(nlog n) messages, if the ID space is of at most linear size. We also show that Las Vegas algorithms (that never fail) require Θ(n) messages. For the synchronous clique under adversarial wake-up, we show that Ω(n^3/2) is a tight lower bound for randomized 2-round algorithms. Finally, we turn our attention to the asynchronous clique: Assuming adversarial wake-up, we give a randomized algorithm that achieves a message complexity of O(n^1 + 1/k) and an asynchronous time complexity of k+8. 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.
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