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

Abstract

This article 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 Õ( D + √ n ) time and exchanges Õ( m ) messages (both with high probability), where n is the number of nodes of the network, D is the hop-diameter, and m is the number of edges. This is the first distributed MST algorithm that matches simultaneously the time lower bound of Ω ˜ ( D + √ n ) [10] and the message lower bound of Ω ( m ) [31], which both apply to randomized Monte Carlo 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 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 both Ω ˜ ( D + √ n ) rounds and Ω ( m ) messages.