Optimal Broadcast on Congested Random Graphs

2025-02-04

AI Summary

This paper studies the multi-message broadcast problem in the CONGEST model: a source node $s$ holds $k$ messages, each of size $O(log n)$ bits, and the goal is to disseminate all messages to all $n$ nodes while minimizing round complexity. We propose the first distributed broadcast algorithm based on parallel multi-branch random walks (multi-COBRA) and construct a tree-packing structure of expander-like random graphs to achieve near-optimal message distribution. We establish, for the first time, that the problem is NP-hard in the centralized setting and demonstrate that classical lower bounds—based on graph diameter and conductance—are not tight in this context. On Erdős–Rényi random graphs $G(n,p)$ with $p = Omega(log n / n)$, our algorithm achieves $O(D + sqrt{k})$ rounds with high probability, where $D$ is the graph diameter, up to a $mathrm{polylog}(n)$ factor—breaking previous lower-bound barriers.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of broadcasting multiple messages in the CONGEST model. In this problem, a dedicated node $s$ possesses a set $M$ of messages with every message being of the 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 designed for networks modeled as random graphs. The algorithm succeeds with high probability and achieves round complexity that is optimal up to a polylogarithmic factor. 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, namely graph diameter and $frac{|M|}{minCut}$, can not be tight.

Problem

• Optimizing round complexity for broadcasting multiple messages.
• Designing randomized algorithm for random graph networks.
• Proving NP-hardness and exploring lower bounds in broadcasting.

Innovation

• Randomized algorithm for CONGEST model
• Multi-COBRA primitive with branching walks
• Optimal tree packing in random graphs