Self-Stabilizing Leader Election in Regular Graphs

Population protocols [3] are used as a distributed model that captures the behavior of passively mobile agents. Leader election is one of the most well-studied problems in this model. In this paper, we focus on the self-stabilizing leader election (SSLE) problem proposed by Angluin et al. [5]. Previously, it is known that SSLE can be performed on arbitrary rings and tori with a constant number of states [11], but SSLE on complete graphs requires Ω(n) states [9]. In this paper, we propose the first SSLE population protocol for arbitrary k-regular graphs, which solves an open question proposed in [5]. There are two different SSLE protocols in this paper. In both protocols, the number of states is independent of the size of the graph. The first protocol is simpler and more intuitive but requires O((64c)k · k4k+4) states, where c is the constant number of states used by the SSLE protocol for rings [11]. The second protocol is more carefully designed to reduce the number of states to O(k12). Both of these two constructions can apply to arbitrary graphs if every node knows its own degree.