Minimalist Leader Election Under Weak Communication

TL;DR

This work tackles Leader Election in ultra-sparse distributed models (beeping and stone-age) under minimal assumptions by introducing a uniform, six-state protocol (BFW) that does not rely on unique IDs or prior network knowledge. The method uses random beeps and a one-round freezing mechanism to generate outward beep waves, whose interactions deterministically eliminate leaders until one remains; an Ohm-like flow analysis underpins the correctness, with a probabilistic argument ensuring convergence whp. The main results prove almost-sure convergence with a high-probability time bound of $O(D^2 \log n)$, with a faster $O(D \log n)$ bound achievable when a constant-factor approximation of the diameter is available. The work highlights a favorable trade-off between extreme simplicity and performance, making it appealing for simple biological or nano-scale systems, while acknowledging limitations such as lack of termination detection and the overhead tied to diameter. Overall, it advances the understanding of what can be computed in the beeping/stone-age regime with minimal memory and symmetry-breaking resources, and it frames clear avenues for tightening bounds and exploring robustness.

Abstract

We propose a protocol to solve Leader Election within weak communication models such as the beeping model or the stone-age model. Unlike most previous work, our algorithm operates on only six states, does not require unique identifiers, and assumes no prior knowledge of the network's size or topology, i.e., it is uniform. We show that under our protocol, the system almost surely converges to a configuration in which a single node is in a leader state. With high probability, this occurs in fewer than $O(D^2 \log n)$ rounds, where $D$ is the network diameter. We also show that this can be decreased to $O(D \log n)$ when a constant factor approximation of $D$ is known. The main drawbacks of our approach are a $\TildeΩ(D)$ overhead in the running time compared to algorithms with stronger requirements, and the fact that nodes are unaware of when a single-leader configuration is reached. Nevertheless, the minimal assumptions and natural appeal of our solution make it particularly well-suited for implementation in the simplest distributed systems, especially biological ones.

Figures (1)

• Figure 1: Definition of Algorithm BFW as a probabilistic finite-state machine. The starting state is $W^\bullet$. Solid lines indicate transitions corresponding to $\delta_\top$, which occur when a beep is heard, while dashed lines indicate transitions corresponding to $\delta_\bot$, which occur when both the node and its neighborhood are listening. A node beeps if its state belongs to $Q_b = \{B^\bullet, B^\circ\}$, which are circled twice in the figure. A node is considered as a leader if its state belongs to $\{B^\bullet, F^\bullet, W^\bullet\}$, which make up the first half of the figure. All transitions are deterministic except for $\delta_\bot(W^\bullet)$.

Theorems & Definitions (38)

• Definition 1: Eventual Leader Election
• Theorem 2
• Theorem 3
• Definition 4: Paths
• Definition 5: Flow
• Claim 6: Basic Observations
• proof
• Lemma 7: Conservation of flow
• proof
• Corollary 8: Ohm's law