On Space and Time Complexity of Loosely-Stabilizing Leader Election

Loose stabilization is a relaxed notion of self-stabilization, which guarantees algorithms to converge and keep some desired behavior from any initial configuration, but allows the algorithms to drop out of it after a sufficiently long period. In this paper, we investigate the complexity of the loosely-stabilizing leader election problem in the population protocol model under the probabilistic scheduler. The primary contribution is to give lower bounds for the expected length of convergence periods and the memory space. Precisely, for any loosely-stabilizing leader election algorithm with stabilization periods of length $\Omega\expN$ in expectation, each agent needs ΩlogN-bit memory space, and the expected convergence length is ΩNn, where n is the unknown number of agents, and N is the upper bound knowledge for n available to the algorithm. We also show the matching upper bounds by proposing a new loosely-stabilizing leader election algorithm, which slightly improves the expected convergence length of the previously known algorithm by Sudo et al. [15] without any degradation of the memory usage or the stabilization length.