The Same Speed Timer in Population Protocols

Abstract

A novel concept of the same speed timer is presented, and is applied in the population protocol (PP) model to improve the convergence time of existing loosely-stabilizing leader election protocols. Loosely-stabilizing leader election guarantees that, starting from any configuration, the system reaches a safe configuration within a short time (convergence), and after that, the system keeps the unique leader for a long time (closure). Two loosely-stabilizing leader election protocols for arbitrary graphs exist in the literature, one uses identifiers of nodes and the other uses random numbers to elect a unique leader. Both protocols guarantee that the expected convergence time is polynomial and the expected holding time (the time the leader is kept) is exponential. In this paper, convergence time of these protocols is dramatically improved by the same speed timer without impairing the exponential holding time. Specifically, a fast deterministic loosely-stabilizing leader election protocol that uses identifiers of nodes and a fast randomized loosely-stabilizing leader election protocol are given. The expected convergence time and expected holding time of the former protocol are O(mN log N) and Ω(Ne2N), respectively, where m is the number of edges in the graph and N is a given upper bound on the number of nodes n. The expected convergence time and expected holding time of the latter protocol are O(mN2 log n) and Ω(Ne2N), respectively. A self-stabilizing two-hop coloring protocol that uses only O(log n) memory space of each agent is given as a tool of the latter protocol. A lower bound is also given: any loosely-stabilizing leader election protocol with expected exponential holding time requires Ω(mN) expected convergence time.