Broadcast in Almost Mixing Time

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.