The Complexity of Leader Election: A Chasm at Diameter Two

TL;DR

This paper determines the exact message complexity for implicit leader election in diameter-two networks, completing the understanding of how network diameter influences the complexity of leader election algorithms.

Contribution

It establishes a tight bound of tenilde;tenilde;tenilde;(n) for diameter-two networks, filling a key gap in the complexity characterization based on graph diameter.

Findings

Message complexity for diameter-two networks is tenilde;tenilde;tenilde;(n).

Complexity bounds are tight and match the known bounds for diameters one and three.

The results fully characterize leader election complexity relative to network diameter.

Abstract

This paper focuses on studying the message complexity of implicit leader election in synchronous distributed networks of diameter two. Kutten et al.\ [JACM 2015] showed a fundamental lower bound of $\Omega(m)$ ($m$ is the number of edges in the network) on the message complexity of (implicit) leader election that applied also to Monte Carlo randomized algorithms with constant success probability; this lower bound applies for graphs that have diameter at least three. On the other hand, for complete graphs (i.e., graphs with diameter one), Kutten et al.\ [TCS 2015] established a tight bound of $\tilde{\Theta}(\sqrt{n})$ on the message complexity of randomized leader election ($n$ is the number of nodes in the network). For graphs of diameter two, the complexity was not known. In this paper, we settle this complexity by showing a tight bound of $\tilde{\Theta}(n)$ on the message complexity of leader election in diameter-two networks. Together with the two previous results of Kutten et al., our results fully characterize the message complexity of leader election vis-\`a-vis the graph diameter.