# An Almost Singularly Optimal Asynchronous Distributed MST Algorithm

A singularly (near) optimal distributed algorithm is one that is (near) optimal in two criteria, namely, its time and message complexities. For synchronous CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general asynchronous CONGEST networks. We present a randomized distributed MST algorithm that, with high probability, computes an MST in asynchronous CONGEST networks and takes Õ(D^1+ϵ + √(n)) time and Õ(m) messages, where n is the number of nodes, m the number of edges, D is the diameter of the network, and ϵ >0 is an arbitrarily small constant (both time and message bounds hold with high probability). Our algorithm is message optimal (up to a polylog(n) factor) and almost time optimal (except for a D^ϵ factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all D = O(n^1-ϵ)) and uses Õ(m) messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the KT_1 CONGEST model. A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth Õ(D^1+ϵ) (for an arbitrarily small constant ϵ > 0) in Õ(D^1+ϵ) time and Õ(m) messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting.