Silent Self-Stabilizing Ranking: Time Optimal and Space Efficient

TL;DR

This work tackles ranking in the population protocol model by designing a silent, self-stabilizing protocol that assigns a unique rank to each of $n$ anonymous agents, thereby enabling leader election. It introduces SpaceEfficientRanking, a non-self-stabilizing baseline with $n + \Theta(\log n)$ overhead that stabilizes in $O(n^2 \log n)$ interactions w.h.p., and Ranking+, a self-stabilizing extension with $n + O(\log^2 n)$ overhead that remains silent and stabilizes in the same time bound. The core ideas combine a space-efficient phased ranking with one-way epidemics and a reset-propagation mechanism to recover from arbitrary faults, achieving an exponential improvement in overhead over previous self-stabilizing approaches. The results establish time-optimality within the silent class and open questions about further reducing overhead or achieving deterministic stabilization with subexponential states. Practical impact lies in robustly coordinating large anonymous populations with limited memory using simple pairwise interactions.

Abstract

We present a silent, self-stabilizing ranking protocol for the population protocol model of distributed computing, where agents interact in randomly chosen pairs to solve a common task. We are given $n$ anonymous agents, and the goal is to assign each agent a unique rank in $\{1, \dots, n\}$. Given unique ranks, it is straightforward to select a designated leader. Thus, our protocol is a self-stabilizing leader election protocol as well. Ranking requires at least $n$ states per agent; hence, the goal is to minimize the additional number of states, called overhead states. The core of our protocol is a space-efficient but non-self-stabilizing ranking protocol that requires only $n + O(\log n)$ states. Our protocol stabilizes in $O(n^2\log n)$ interactions w.h.p.\ and in expectation, using $n + O(\log^2 n)$ states in total. Our stabilization time is asymptotically optimal (see Burman et al., PODC'21). In comparison to the currently best known ranking protocol by Burman et al., which requires $n + Ω(n)$ states, our result exponentially improves the number of overhead states.

Figures (3)

• Figure 1: A high-level overview of the self-stabilizing algorithm.
• Figure 2: Number of ranked agents (blue), and average of the phase counters stored by unranked agents (red, dashed), as a function of the number of interactions. The protocol (for $n=256$) is initialized as follows: 255 agents are ranked (with ranks $2, \ldots, 256$), and one agent is a phase agent with maximum liveness counter.
• Figure 3: Number of interactions (normalized by $n^2$) needed to reach a state in which $c \cdot n$ agents are ranked, starting from the following configuration: there is one agent in rank 1 (the unaware leader), and all other agents are still in a leader election state. We performed 100 simulations per value of $n \in \{2^i \mid i \in {\mathbb{N}}, 7 \leq i \leq 13\}$.

Theorems & Definitions (60)

• Theorem 1: label=thm:non-self-stab-ranking,restate=thmOne
• Theorem 2: label=thm:ss-ranking,restate=thmTwo
• Lemma 3
• Lemma 4
• Definition 5
• Lemma 6
• proof
• Lemma 7
• proof
• proof : Proof of \ref{['thm:ss-ranking']}