Broadcast in Almost Mixing Time

TL;DR

This paper tackles multi-message broadcast in the CONGEST model, focusing on expander graphs with small mixing time. It introduces a two-pronged strategy: (i) embed a virtual Erdős--Rényi graph atop the host network to exploit fast tree packing via COBRA (coalescing-branching random walks), and (ii) map the solution back to the original graph through a lifting argument that incurs a factor of $\tau_{mix}$ and polylog terms. The main contributions are a universal algorithm achieving near-optimal round complexity on expanders, a near-optimal $O(\log^2 n + \log n \cdot \frac{k}{\delta(G)})$-style broadcast on Erdős--Rényi graphs via a provably efficient tree packing, and a rigorous NP-hardness result for exact round computation in centralized settings. The work also develops a spectral-graph toolkit to show that regularizing random graphs preserves expansion properties, enabling effective parallel COBRA execution and robust tree packing. These results progress understanding of topology-adaptive broadcasting and provide techniques potentially useful beyond broadcasting, such as distributed tree packing on random graphs and expander-based information dissemination.

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.

Figures (4)

• Figure 1: An example where diameter and minimum cut are not telling. Edge labels denote bandwidth.
• Figure 2: An example of mapping a graph with arbitrary bandwidths to a graph suitable for CONGEST.
• Figure 3: An example graph $G$ where diameter and minimum cut are not telling. Here, edge labels denote bandwidth.
• Figure 4: Example of transforming a layered graph with arbitrary bandwidths into a graph suitable for CONGEST. Here the number of messages $n$ is $3$.

Theorems & Definitions (33)

• Definition 1: Multi-message broadcast
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6: Lemma 2.5 ghaffari2017distributed
• Lemma 7
• Theorem 8: rivera2020lecture
• Theorem 9: Cooper et al. cooper2016coalescing
• Lemma 9