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.

翻译：我们研究了在CONGEST模型中广播多条消息的问题。在此问题中，一个专用的源节点$s$持有一个消息集合$M$，每条消息的大小为$O(\log n)$，其中$n$是节点总数。目标是确保网络中的每个节点都能学习到$M$中的所有消息。算法的执行以轮次进行，我们专注于优化广播多条消息的轮次复杂度。我们的主要贡献是针对具有扩展器拓扑结构的网络提出了一种随机算法，这种拓扑在实践中被广泛用于构建可扩展且鲁棒的分布式系统。该算法以高概率成功，并实现了在轮次复杂度上达到网络混合时间与多对数项乘积的最优性。它利用了多-COBRA原语，该原语使用多个并行运行的分支随机游走。据我们所知，这种方法此前尚未应用于分布式算法中。我们方法的一个关键方面是利用这些分支随机游走来构建随机图的最优（达到多对数因子）树打包，然后将其用于高效广播。这一结果本身具有独立的研究价值。我们还证明了该问题在集中式设置下是NP难的，并深入探讨了为什么针对一般图的简单下界——即图的直径和$\frac{|M|}{\textit{最小割}}$——不可能是紧的。