Singing a MIS

TL;DR

The paper introduces the singing model, a multi-note extension of the beeping model, and studies how oblivious agents can compute a maximal independent set (MIS) without knowledge of network size or topology. It presents two simple, memory-efficient protocols (Singing and Self-Jamming Singing) that drive the network toward an MIS by having agents compete via a bit-driven note scheme; progress is measured by contraction in the number of active edges. The authors prove logarithmic-time convergence (O(log n)) in static, synchronous networks and extend the results to asynchronous rounds and dynamic, fault-prone networks with bounded disruption, providing rigorous probabilistic analyses and lemmas (including the key Lemma 'elig'). These results yield fault-tolerant, network-oblivious MIS protocols applicable to diverse settings (e.g., wireless, biological, and robotic networks) where full topology knowledge is unrealistic. The work also contrasts with prior models (beeping, radio, Stone Age) by achieving logarithmic convergence under dynamic conditions and constraints on agent memory and information propagation.

Abstract

We introduce a broadcast model called the singing model, where agents are oblivious of the size and structure of the communication network, even their immediate neighborhood. Agents can sing multiple notes which are heard by their neighbors. The model is a generalization of the beeping model, where agents can only emit sound at a single frequency. We give a simple and natural protocol where agents compete with their neighbors and their strength is reflected in the number of notes they sing. It converges in $O(log(n))$ time with high probability, where $n$ is the number of agents in the network. The protocol works in an asynchronous model where rounds vary in length and have different start times. It works with completely dynamic networks where agents can be faulty. The protocol is the first to converge to an MIS in logarithmic time for dynamic networks in a network oblivious model.

Figures (5)

• Figure 1: Singing protocol
• Figure 2: Transition logic for the singing protocol. The label $0$ on an arrow indicates that the agent hears note $0$, and $\neg 0$ indicates that the agent does not hear note $0$ in the round. Recall that note $0$ is heard if and only if the agent has a neighbor in the $\textrm{\sc In}$ state. The label $>$ on an arrow indicates that the $\ell$-value for the agent is strictly larger than that of all of its $\textrm{\sc Un}$ neighbors, and $\le$ indicates that the $\ell$-value for the agent is less than or equal to the $\ell$-value of at least one of its $\textrm{\sc Un}$ neighbors. The dotted lines show transitions that will not arise in the case of a static network.
• Figure 3: Transition logic for the singing self-jamming protocol. The label $0$ on an arrow indicates that the agent hears note $0$, and $\neg 0$ indicates that the agent does not hear note $0$ in the round. Recall an agent sings note $0$ if and only if the agent is in the $\textrm{\sc In}$ state. For $\textrm{\sc In}$ agents, an outgoing arrow labeled $<$ indicates that its $\ell$-value is strictly less than the $\ell$-value of one of its $\textrm{\sc In}$ neighbors. An outgoing arrow labeled $\ge$ indicates that its $\ell$-value is greater than or equal to the $\ell$-values of all of its $\textrm{\sc In}$ neighbors. For $\textrm{\sc Un}$ agents, an outgoing arrow labeled $<$ indicates that its $\ell$-value is strictly less than the $\ell$-value of one of its $\textrm{\sc Un}$ neighbors. An outgoing arrow labeled $\ge$ indicates that its $\ell$-value is greater than or equal to the $\ell$-values of all of its $\textrm{\sc Un}$ neighbors.
• Figure 4: In the network on the left, if the edge $\{u,v\}$ is added or deleted at the beginning of round $i$ or if $u$ or $v$ execute their protocol incorrectly at the end of round $i-1$, then the edge $\{u,v\}$ is said to be changed at the beginning of round $i$. In this case, the edges $\{v,x\}$, $\{x,y\}$, and $\{y,z\}$ are affected by the change at the beginning of round $i$. In the network on the right, if the edge $\{u,v\}$ is not affected by a change at the beginning of round $i$, then all of the agents that are a distance at most $3$ from $u$ or $v$ execute their protocol correctly at the end of round $i-1$ and the network induced by those agents is unchanged at the beginning of round $i$.
• Figure 5: Self-jamming singing protocol - round $(v,i)$

Theorems & Definitions (51)

• Definition 3.1
• Claim 4.1
• proof
• Claim 4.2
• proof
• Lemma 4.3
• proof
• Definition 4.4: Changed and Affected Agents and Edges
• Claim 4.5
• Lemma 4.6