Loosely-Stabilizing Leader Election on Arbitrary Graphs in Population Protocols

Abstract

We consider the leader election problem in population protocol models. In pragmatic settings of population protocols, self-stabilization is a highly desired feature owing to its fault resilience and the benefit of initialization freedom. However, the design of self-stabilizing leader election is possible only under a strong assumption (i.e., the knowledge of the exact size of a network) and rich computational resource (i.e., the number of states). Loose-stabilization, introduced by Sudo et al. [Theoretical Computer Science, 2012], is a promising relaxed concept of selfstabilization to address the aforementioned issue. Loose-stabilization guarantees that starting from any configuration, the network will reach a safe configuration where a single leader exists within a short time, and thereafter it will maintain the single leader for a long time, but not forever. The main contribution of the paper is a time-optimal loosely-stabilizing leader election protocol. While the shortest convergence time achieved so far in loosely-stabilizing leader election is O(log 3 n) parallel time, the proposed protocol with design parameter τ ≥ 1 attains O(τ log n) parallel convergence time and Ω(n τ ) parallel holding time (i.e., the length of the period keeping the unique leader), both in expectation. This protocol is time-optimal in the sense of both the convergence and holding times in expectation because any loosely-stabilizing leader election protocol with the same length of the holding time is known to require Ω(τ log n) parallel time.

Loosely-stabilizing leader election with polylogarithmic convergence time

“…The basic idea of the virus-war mechanism is first presented in [15]. P PL uses this idea, but implements it in a considerably different way in order to reduce the number of leaders to one within a polylogarithmic parallel time.…”

“…Recently, Izumi [11] give a method which improves the convergence time of this protocol to linear time, i.e., O(N ). In [15,16,17], LS-LE protocols are presented for a population where some pairs of agents may not have interactions, i.e., the interaction graph is not complete.…”

Loosely Stabilizing Leader Election on Arbitrary Graphs in Population Protocols without Identifiers or Random Numbers

“…In this section, we give a loosely stabilizing leader election † Lemma 15 in [23] claims Pr(RT Γ (i) < 2im ln n ) ≥ 1 − ne −i/4 for any i ≥ 1. We obtain Pr(RT Γ (i) < im(1 + ln n)) ≥ 1 − ne −i/4 replacing 2im ln n with im(1+ln n) in all the inequalities that appear in the proof of Lemma 15 in [23].…”

“…Each of the five conditions (A), (B), (C), (D), and (E) holds with probability at least 1−O(nδ log n•e −τ ), as claimed by Lemmas 4, 5, 6, 7, and 8, respectively. Fortunately, these lemmas can be proven by a simple application of Chernoff bounds and/or by techniques used in [23]. See Appendix for the proofs of these lemmas.…”

“…Proof : This lemma follows from the proof of Lemma 5 in [23]. That proof claims that if there is a leader in a configuration, then the timer L of every agent is more than t max /2 in the configuration reached by the next 2mτ interactions with probability at least 1 − 2ne τ .…”