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.

我们提供了第一个异步分布式算法，用于在具有n个节点和m个边的足够密集的图中计算具有o（m）位通信的广播和最小生成树。几十年来，人们认为，构成广播树的任何算法都需要\（\ varOmega（m）\）位通信。在2015年，King，Kutten和Thorup表明，在KT1模型中，节点对邻居的身份有了初步了解，可以在同步CONGEST的\（{\ tilde {O}}（n）\）消息中构造MST。模型。在CONGEST模型中，消息的大小为\（O（\ log n）\）。但是，没有算法使用o（m）条消息在异步情况下是已知的。在这里，我们提供了一种算法，该算法使用\（O（n ^ {3/2} \ log ^ {3/2} n）\）消息在异步CONGEST模型中查找MST。我们的算法是随机蒙特卡洛算法，并以高概率输出MST。我们将提供一种算法，用于使用\（O（n ^ {3/2} \ log ^ {3/2} n）\）条消息来计算生成树。给定生成树，我们可以使用\（{\ tilde {O}}（n）\）消息来计算MST 。