Beeping Deterministic CONGEST Algorithms in Graphs

2025-02-19

AI Summary

This work addresses the problem of deterministic simulation of CONGEST algorithms in the Beeping Network (BN) model, aiming to overcome bandwidth limitations and interference avoidance under minimal signaling. The proposed methodology introduces a hierarchical pipelined *h*-hop simulation framework, graph-structure-aware beep sequence design, non-adaptive and adaptive distributed coding schemes, a deterministic message scheduling mechanism, and integrates efficient network decomposition with hierarchical information aggregation. The contributions include: (i) the first deterministic CONGEST simulation in *O*(Δ² logᴼ(¹) *n*) rounds—improving upon the prior Θ(Δ³) bound; (ii) reducing the deterministic time complexity of computing a maximal independent set (MIS) in BN from Θ(Δ³) to *O*(Δ² logᴼ(¹) *n*); and (iii) establishing a tight Ω(Δʰ⁺¹) lower bound for *h*-hop simulation and designing a near-optimal *h*-hop simulation algorithm matching this bound asymptotically.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Sampling/Simulation-based Search
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior

Application Category

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

Abstract

The Beeping Network (BN) model captures important properties of biological processes. Paradoxically, the extremely limited communication capabilities of such nodes has helped BN become one of the fundamental models for networks. Since in each round, a node may transmit at most one bit, it is useful to treat the communications in the network as distributed coding and design it to overcome the interference. We study both non-adaptive and adaptive codes. Some communication and graph problems already studied in BN admit fast randomized algorithms. On the other hand, all known deterministic algorithms for non-trivial problems have time complexity at least polynomial in the maximum node-degree $Delta$. We improve known results for deterministic algorithms showing that beeping out a single round of any congest algorithm in any network can be done in $O(Delta^2 log^{O(1)} n)$ beeping rounds, even if the nodes intend to send different messages to different neighbors. This upper bound reduces polynomially the time for a deterministic simulation of congest in a BN, comparing to the best known algorithms, and nearly matches the time obtained recently using. Our simulator allows us to implement any efficient algorithm designed for the congest networks in BN, with $O(Delta^2 log^{O(1)} n)$ overhead. This $O(Delta^2 log^{O(1)} n)$ implementation results in a polynomial improvement upon the best-to-date $Theta(Delta^3)$-round beeping MIS algorithm. Using a more specialized transformer and some additional machinery, we constructed various other efficient deterministic Beeping algorithms for other commonly used building blocks, such as Network Decomposition. For $h$-hop simulations, we prove a lower bound $Omega(Delta^{h+1})$, and we design a nearly matching algorithm that is able to ``pipeline'' the information in a faster way than working layer by layer.

Problem

• Improve deterministic algorithms in Beeping Networks
• Reduce time complexity for congest algorithm simulation
• Design efficient deterministic Beeping algorithms for network problems

Innovation

• Deterministic CONGEST algorithm simulation
• Beeping Network with O(Δ² log^O(1) n) overhead
• Specialized transformer for efficient algorithms