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.
Categories: cs.DC cs.DS