We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with o(m) bits of communication, in a sufficiently dense graph with n nodes and m edges. For decades, it was believed that Ω(m) bits of communication are required for any algorithm that constructs a broadcast tree. In 2015, King, Kutten and Thorup showed that in the KT1 model where nodes have initial knowledge of their neighbors' identities it is possible to construct MST inÕ(n) messages in the synchronous CONGEST model. In the CONGEST model messages are of size O(log n). However, no algorithm with o(m) messages were known for the asynchronous case. Here, we provide an algorithm that uses O(n 3/2 log 3/2 n) messages to find MST in the asynchronous CONGEST model. Our algorithm is randomized Monte Carlo and outputs MST with high probability. We will provide an algorithm for computing a spanning tree with O(n 3/2 log 3/2 n) messages. Given a spanning tree, we can compute MST withÕ(n) messages.

This paper concerns the problem of constructing a minimum spanning tree (MST) in a synchronous distributed network with n nodes, where each node knows only the identities of itself and its neighbors. We assume the CONGEST model where messages are of size O(log n) bits. Spanning tree construction was long believed to require an amount of communication linear in the number of edges. In 2015, King, Kutten and Thorup presented a Monte Carlo algorithm which broke this communication bound. In particular it showed that an MST could be constructed with time and message complexity O(n log 2 n/ log log n), independent of the number of edges. Here we give trade-offs between time and communication. Our Monte Carlo algorithm runs in O(n/) time and O(n 1+ log log n) messages for any 1 > ≥ log log n/ log n. For the spanning tree problem, we show a time bound of O(n) and a communication bound of O(n log n log log n) messages. We also provide the first algorithm that constructs an MST in time proportional to the diameter of the MST up to a logarithmic factor with o(m) communication.

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