Time-Optimal Self-Stabilizing Leader Election in Population Protocols

Abstract

We consider the standard population protocol model, where (a priori) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time Θ( 2 ) and has the optimal number of states in a population of agents. The existing protocol has the additional property that it becomes silent, i.e., the agents' states eventually stop changing.Observing that any silent protocol solving self-stabilizing leader election requires Ω( ) expected parallel time, we introduce a silent protocol that uses optimal ( ) parallel time and states. Without any silence constraints, we show that it is possible to solve self-stabilizing leader election in asymptotically optimal expected parallel time of (log ), but using at least exponential states (a quasipolynomial number of bits). All of our protocols (and also that of Cai et al.) work by solving the more difficult ranking problem: assigning agents the ranks 1, . . . , .

A Combinatorial Characterization of Self-Stabilizing Population Protocols

“…Beauquier, Blanchard, and Burman extend this work by presenting self-stabilizing leader election in arbitrary graphs when a composition of eventual leader detectors is introduced into the model [6]. Knowledge of the number of agents allows Burman, Doty, Nowak, Severson, and Xu to develop several efficient self-stabilizing protocols for leader election; with no silence or space constraints they achieve optimal expected parallel time of Oplog nq [9]. Loosely-stabilizing protocols relax self-stabilization to allow more tractable solutions, such as leader election protocols with polylogarithmic convergence time by Sudo, Ooshita, Kakugawa, Masuzawa, Datta, and Larmore [19].…”

Efficient Assignment of Identities in Anonymous Populations

“…The labeling problem has been also studied in the context of self-stabilizing protocols where the agents start in arbitrary (not predefined) states, see [16,17]. In [17], Cai et al propose a solution which coincides with our first example of labeling protocols presented in the introduction.…”

“…In [17], Cai et al propose a solution which coincides with our first example of labeling protocols presented in the introduction. In a very recent work [16], Burman et al study both slow and fast labeling protocols focusing mainly on the asymptotic bounds, and with the latter utilizing exponential number of states. The protocols in both papers require the exact knowledge of n and they are not safe.…”

“…We say that a labeling protocol is safe if, for any given set of agents: (i) there exists a finite run of the protocol that produces a valid agent labeling; and (ii) at any time step in any run of the protocol, no two agents have the same label. While the concept of a silent population protocol is well established in the literature [16], the concept of a safe labeling protocol is new. Note that a safe labeling protocol is partially correct in a strong sense since at any time step of its run, the assigned labels form a valid partial labeling.…”

“…Observe that among the three examples of labeling protocols, only the second one is both silent and safe. The first example protocol is silent [16] but not safe. Finally, the third (probabilistic) one is silent and almost safe as it violates the condition (ii) only with small probability.…”