How to Elect a Leader Faster than a Tournament

TL;DR

This paper presents new algorithms that elect a leader faster than traditional tournaments in asynchronous systems, achieving $O( ext{log}^* n)$ time with $O(n^2)$ messages, and applies these results to improve renaming problem bounds.

Contribution

It introduces algorithms that reduce leader election time to $O( ext{log}^* n)$ in asynchronous systems, surpassing the longstanding $ heta( ext{log} n)$ bound, and establishes tight message complexity bounds.

Findings

Leader election time improved to $O( ext{log}^* n)$

Achieves $O(n^2)$ message complexity for leader election and renaming

Proves $ ext{Omega}(n^2)$ message lower bound, closing complexity gaps

Abstract

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 $\Theta(\log n)$-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.