How to Elect a Leader Faster than a Tournament

The problem of electing a leader from among n contenders is one of the fundamental questions in distributed computing. In its simplest formulation, the task is as follows: given n processors, all participants must eventually return a win or lose indication, such that a single contender may win. Despite a considerable amount of work on leader election, the following question is still open: can we elect a leader in an asynchronous fault-prone system faster than just running a (log n)-time tournament, against a strong adaptive adversary?

In this paper, we answer this question in the affirmative, improving on a decades-old upper bound. We introduce two new algorithmic ideas to reduce the time complexity of electing a leader to O(log n), using O(n^2) point-to-point messages. A non-trivial application of our algorithm is a new upper bound for the tight renaming problem, assigning n items to the n participants in expected O(log^2 n) time and O(n^2) messages. We complement our results with lower bound of Omega(n^2) messages for solving these two problems, closing the question of their message complexity.