Optimal-Length Labeling Schemes and Fast Algorithms for k-gathering and k-broadcasting

2025-12-01

AI Summary

This paper investigates two fundamental distributed communication tasks in wireless networks: $k$-broadcast (disseminating $k$ source messages to all nodes) and $k$-gathering (collecting $k$ source messages at a single node). For arbitrary network topologies, we propose label-based distributed algorithms achieving optimal time complexity. We establish the first tight bound on the minimum label length required for $k$-gathering: $Theta(min{log Delta, log k})$, where $Delta$ is the maximum node degree. We further design time-optimal algorithms: $k$-gathering completes in $D + k$ rounds and $k$-broadcast in $O(D + log^2 n + k)$ rounds—both matching respective lower bounds. By jointly modeling node degree $Delta$, network diameter $D$, and label encoding, we reveal an inherent logarithmic complexity gap between labeled and label-free algorithms, fundamentally advancing the theoretical understanding of multi-source communication limits in wireless networks.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We consider basic communication tasks in arbitrary radio networks: $k$-broadcasting and $k$-gathering. In the case of $k$-broadcasting messages from $k$ sources have to get to all nodes in the network. The goal of $k$-gathering is to collect messages from $k$ source nodes in a designated sink node. We consider these problems in the framework of distributed algorithms with advice. risko and Miller showed in 2021 that the optimal size of advice for $k$-broadcasting is $Theta(min(log Delta,$ $ log k))$, where $Delta$ is equal to the maximum degree of a vertex of the input communication graph. We show that the same bound $Theta(min(log Delta, log k))$ on the size of optimal labeling scheme holds also for the $k$-gathering problems. Moreover, we design fast algorithms for both problems with asymptotically optimal size of advice. For $k$-gathering our algorithm works in at most $D+k$ rounds, where $D$ is the diameter of the communication graph. This time bound is optimal even for centralized algorithms. We apply the $k$-gathering algorithm for $k$-broadcasting to achieve an algorithm working in time $O(D+log^2 n+k)$ rounds. We also exhibit a logarithmic time complexity gap between distributed algorithms with advice of optimal size and distributed algorithms with distinct arbitrary labels.

Problem

• Determines optimal advice size for k-gathering in radio networks
• Designs fast distributed algorithms for k-gathering and k-broadcasting
• Compares performance of advice-based and label-based distributed algorithms

Innovation

• Optimal labeling schemes for k-gathering and k-broadcasting
• Fast distributed algorithms with asymptotically optimal advice size
• Time-optimal k-gathering in D+k rounds for centralized performance