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

TL;DR

This paper introduces the first asynchronous distributed algorithms for broadcast and minimum spanning tree construction that operate with o(m) message complexity, challenging the long-held belief that Ω(m) messages are necessary.

Contribution

It presents the first asynchronous algorithms achieving o(m) message complexity for broadcast and MST, with a randomized approach that works with high probability.

Findings

MST can be computed with O(n^{3/2} log^{3/2} n) messages in asynchronous models.

Given a spanning tree, MST can be derived with ~O(n) messages.

The algorithms operate with high probability and are the first of their kind in asynchronous settings.

Abstract

We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with $o(m)$ bits of communication, in a graph with $n$ nodes and $m$ edges. For decades, it was believed that $\Omega(m)$ 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 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 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 with $\tilde{O}(n)$ messages.