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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 + &sqrt; 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 + &sqrt; 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 + &sqrt; n) rounds and Ω (m) messages.

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