Broadcast in Almost Mixing Time

TL;DR

This paper presents a randomized broadcasting algorithm for expander networks in the CONGEST model, achieving near-optimal round complexity by using novel multi-branching random walks and tree packings.

Contribution

Introduces a new multi-COBRA primitive with branching random walks for efficient broadcasting in expander graphs, a novel approach in distributed algorithms.

Findings

Achieves near-optimal round complexity up to mixing time and polylogarithmic factors.

Uses multi-branching random walks to construct near-optimal tree packings.

Proves NP-hardness of the problem in centralized settings.

Abstract

We study the problem of broadcasting multiple messages in the CONGEST model. In this problem, a dedicated source node $s$ possesses a set $M$ of messages with every message of size $O(\log n)$ where $n$ is the total number of nodes. The objective is to ensure that every node in the network learns all messages in $M$. The execution of an algorithm progresses in rounds, and we focus on optimizing the round complexity of broadcasting multiple messages. Our primary contribution is a randomized algorithm for networks with expander topology, which are widely used in practice for building scalable and robust distributed systems. The algorithm succeeds with high probability and achieves a round complexity that is optimal up to a factor of the network's mixing time and polylogarithmic terms. It leverages a multi-COBRA primitive, which uses multiple branching random walks running in parallel. To the best of our knowledge, this approach has not been applied in distributed algorithms before. A crucial aspect of our method is the use of these branching random walks to construct an optimal (up to a polylogarithmic factor) tree packing of a random graph, which is then used for efficient broadcasting. This result is of independent interest. We also prove the problem to be NP-hard in a centralized setting and provide insights into why straightforward lower bounds for general graphs, namely graph diameter and $\frac{|M|}{\textit{minCut}}$, cannot be tight.