Improving Efficiency in Near-State and State-Optimal Self-Stabilising Leader Election Population Protocols

TL;DR

The paper tackles the ranking problem in self-stabilising population protocols, aiming silent, stable leader election with subquadratic stabilisation times. It develops three main strategies: (i) a state-optimal, ring-of-traps construction achieving $O(\min(kn^{3/2},n^2\log^2 n))$ whp; (ii) a one-extra-state scheme with lines of traps yielding $O(n^{7/4}\log^2 n)$ whp; and (iii) an $O(\log n)$-extras scheme using perfectly balanced binary trees to reach $O(n\log n)$ whp. Central to these results are agent traps, lines, and carefully designed routing and epidemic mechanisms, underpinned by Chernoff bounds to guarantee high-probability progress. The work significantly narrows the gap toward subquadratic self-stabilising ranking and thus improves practical leader election in distributed populations, while leaving open the existence of purely rank-state protocols with o($n^2$) time. Overall, these contributions advance our understanding of time-space trade-offs in self-stabilising population protocols and their applicability to robust leader election.

Abstract

We investigate leader election problem via ranking within self-stabilising population protocols. In this scenario, the agent's state space comprises $n$ rank states and $x$ extra states. The initial configuration of $n$ agents consists of arbitrary arrangements of rank and extra states, with the objective of self-ranking. Specifically, each agent is tasked with stabilising in a unique rank state silently, implying that after stabilisation, each agent remains in its designated state indefinitely. In this paper, we present several new self-stabilising ranking protocols, greatly enriching our comprehension of these intricate problems. All protocols ensure self-stabilisation time with high probability (whp), defined as $1-n^{-η},$ for a constant $η>0.$ We delve into three scenarios, from which we derive stable (always correct), either state-optimal or almost state-optimal, silent ranking protocols that self-stabilise within a time frame of $o(n^2)$ whp, including: - Utilising a novel concept of an agent trap, we derive a state-optimal ranking protocol that achieves self-stabilisation in time $O(min(kn^{3/2},n^2\log^2 n)),$ for any $k$-distant starting configuration. - Furthermore, we show that the incorporation of a single extra state ($x=1$) ensures a ranking protocol that self-stabilises in time $O(n^{7/4}\log^2 n)=o(n^2)$, regardless of the initial configuration. - Lastly, we show that extra $x=O(\log n)$ states admit self-stabilising ranking with the best currently known stabilisation time $O(n\log n)$, when whp and $x=O(\log n)$ guarantees are imposed.

Figures (2)

• Figure 1: Graph $G$ for $m^2=16$. For example, for $l=1$ we get $l_0=2, l_1=3,$ and $l_2=8$.
• Figure 2: State distribution in perfectly balanced binary tree for $n=9$.

Theorems & Definitions (49)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• Lemma 5