Broadcast and minimum spanning tree with o ( m ) messages in the asynchronous CONGEST model

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 $$\varOmega (m)$$ 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 neighbours’ identities it is possible to construct MST in $${\tilde{O}}(n)$$ messages in the synchronous CONGEST model. In the CONGEST model messages are of size $$O(\log n)$$ . However, no algorithm with o(m) messages was 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 with $${\tilde{O}}(n)$$ messages.