A Time- and Message-Optimal Distributed Algorithm for Minimum Spanning Trees

TL;DR

This paper introduces a randomized distributed algorithm for minimum spanning trees that is optimal in both time and message complexity, matching known lower bounds and improving efficiency in weighted networks.

Contribution

It presents the first distributed MST algorithm that simultaneously matches the optimal time and message lower bounds, with a new lower bound construction for such algorithms.

Findings

Algorithm runs in  O(D + \u221A n) time and  O(m) messages.

Achieves optimal bounds up to polylogarithmic factors for weighted networks.

Provides a new lower bound graph construction requiring both optimal time and message complexity.

Abstract

This paper presents a randomized Las Vegas distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in $\tilde{O}(D + \sqrt{n})$ time and exchanges $\tilde{O}(m)$ messages (both with high probability), where $n$ is the number of nodes of the network, $D$ is the diameter, and $m$ is the number of edges. This is the first distributed MST algorithm that matches \emph{simultaneously} the time lower bound of $\tilde{\Omega}(D + \sqrt{n})$ [Elkin, SIAM J. Comput. 2006] and the message lower bound of $\Omega(m)$ [Kutten et al., J.ACM 2015] (which both apply to randomized algorithms). The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound {\em does not} work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires \emph{both} $\tilde{\Omega}(D + \sqrt{n})$ rounds and $\Omega(m)$ messages.