Self-Stabilizing MIS Computation in the Beeping Model

TL;DR

This work studies self-stabilizing maximal independent set computation in the weak full-duplex beeping model. It introduces a level-based adaptive beeping scheme that eliminates the need for two-round phases and proves fast stabilization guarantees under three knowledge scenarios: a global bound on the maximum degree $\Delta$ yields $O(\log n)$ rounds, per-vertex degree bounds yield $O(\log n\cdot\log\log n)$ rounds, and a two-channel extension with a 1-hop degree bound yields $O(\log n)$. The analysis centers on platinum and golden rounds, developing bounds via auxiliary quantities $\eta_t(v)$ and $\eta'_t(v)$ to ensure progress toward a stable MIS despite arbitrary faults. The results highlight how limited topology information and extra signaling channels can substantially accelerate self-stabilizing MIS in extremely weak communication models, while also leaving open the challenge of topology-free stabilization. $${}$${}

Abstract

We consider self-stabilizing algorithms to compute a Maximal Independent Set (MIS) in the extremely weak beeping communication model. The model consists of an anonymous network with synchronous rounds. In each round, each vertex can optionally transmit a signal to all its neighbors (beep). After the transmission of a signal, each vertex can only differentiate between no signal received, or at least one signal received. We also consider an extension of this model where vertices can transmit signals through two distinguishable beeping channels. We assume that vertices have some knowledge about the topology of the network. We revisit the not self-stabilizing algorithm proposed by Jeavons, Scott, and Xu (2013), which computes an MIS in the beeping model. We enhance this algorithm to be self-stabilizing, and explore three different variants, which differ in the knowledge about the topology available to the vertices and the number of beeping channels. In the first variant, every vertex knows an upper bound on the maximum degree $Δ$ of the graph. For this case, we prove that the proposed self-stabilizing version maintains the same run-time as the original algorithm, i.e., it stabilizes after $O(\log n)$ rounds w.h.p. on any $n$-vertex graph. In the second variant, each vertex only knows an upper bound on its own degree. For this case, we prove that the algorithm stabilizes after $O(\log n\cdot \log \log n)$ rounds on any $n$-vertex graph, w.h.p. In the third variant, we consider the model with two beeping channels, where every vertex knows an upper bound of the maximum degree of the nodes in the $1$-hop neighborhood. We prove that this variant stabilizes w.h.p. after $O(\log n)$ rounds.

Figures (1)

• Figure 1: Beeping probability $p_t(v)$ of $v$ based on value of $\ell_t(v)$.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 5
• Definition 6: Prominent Vertices and Platinum Rounds
• Lemma 7
• Lemma 8: Lower Bound on Platinum Rounds
• Lemma 9: Stopping Times for Platinum Rounds
• Lemma 10