A time- And message-optimal distributed algorithm for minimum spanning trees

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 O (D + √ n) time and exchanges O (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.