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

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 (O) over tilde (D + root n) time and exchanges (O) over tilde (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 simultaneously the time lower bound of (Omega) over tilde (D + root 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 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 (Omega) over tilde (D + root n) rounds and Omega(m) messages.