Beeping Deterministic CONGEST Algorithms in Graphs

TL;DR

This work studies how to efficiently simulate CONGEST-round communication in Beeping Networks (BN), a model with extremely limited communication where nodes may beep or listen. The authors introduce two near-optimal deterministic simulators: a general CONGEST simulator with overhead $O(Δ^2 ext{ polylog } n ext{ log } Δ)$ and a specialized local-broadcast simulator with $O(B Δ^2 ext{ log } n)$ rounds, enabling BN implementations of CONGEST algorithms. They apply the simulators to significantly improve deterministic BN algorithms for core building blocks, including MIS ($O(Δ^2 ext{ polylog } n)$ rounds), Local Broadcast, Learning Neighborhood, Cluster Gathering, and Network Decomposition (all with polynomial overhead in $Δ$ and polylog factors). A multi-hop extension yields $B$-bit $h$-hop simulations with $O(h B Δ^{h+2} ext{ polylog } n)$ rounds and matching lower bounds, highlighting a close alignment between deterministic BN and CONGEST complexities. The results also establish lower bounds and pipeline techniques, providing a foundation for robust, energy-efficient BN algorithms with practical impact in IoT and distributed biological-inspired systems.

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 $Δ$. 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 $O(Δ^2 \log^{O(1)} n)$ 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 $O(Δ^2 \log^{O(1)} n)$ overhead. This $O(Δ^2 \log^{O(1)} n)$ implementation results in a polynomial improvement upon the best-to-date $Θ(Δ^3)$-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 $Ω(Δ^{h+1})$, and we design a nearly matching algorithm that is able to ``pipeline'' the information in a faster way than working layer by layer.

Figures (4)

• Figure 1: Beeping Network communication model example.
• Figure 2: An illustration of the structure of the graph. The graph is partitioned into vertical layers $T_h$, …, $T_1$, $R$. The graph is branching out heavily, so we show only a path from an arbitrary node in $T_1$ layer to an arbitrary node in $T_h$ layer, with all the edges incident to the path. The numbers between the layers denote the number of edges between the layers that are incident to the path or on the path. Recall that layers $T_1$ and $R$ have $\Delta/2$ nodes each, while the other layers have significantly more nodes, but we only show nodes that are adjacent to the considered path.
• Figure 3: Illustration for Theorem \ref{['thm:multihoplb']}. Example of adversarial graph for $\Delta=4$ and $h=5$.
• Figure 4: Illustration of c2b algorithm -- consecutive handshakes between the announcer $c$ and its responders during a phase.

Theorems & Definitions (32)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• Definition 1: Avoiding selectors
• theorem 5: BonisGV05ChlebusK05
• proof
• Lemma 1: Correct receiving
• Lemma 2: Correct realization
• Lemma 3: Sub-phase progress