Move-optimal partial gathering of mobile agents without identifiers or global knowledge in asynchronous unidirectional rings

Abstract

In this paper, we consider the partial gathering problem of mobile agents in asynchronous unidirectional ring networks. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all the k agents distributed in the network terminate at a single node. The partial gathering problem requires, for a given positive integer g(<k), that all the agents terminate in a configuration such that either at least g agents or no agent exists at each node. The requirement for the partial gathering problem is strictly weaker than that for the total gathering problem, and thus it is interesting to clarify the difference on the move complexity between them. In this paper, we aim to solve the partial gathering problem for agents without identifiers or any global knowledge such as the number k of agents or the number n of nodes. We consider deterministic and randomized cases. First, in the deterministic case, we show that the set of unsolvable initial configurations is the same as that for the case of agents with knowledge of k. In addition, we propose an algorithm that solves the problem from any solvable initial configuration in a total number of O(gn) moves. Next, in the randomized case, we propose an algorithm that solves the problem in a total number of O(gn) moves in expectation from any initial configuration. Note that g<k holds and agents require a total number of Ω(gn) (resp., Ω(kn)) moves to solve the partial (resp., total) gathering problem. Thus, our algorithms can solve the partial gathering problem in asymptotically optimal total number of moves without identifiers or global knowledge, and the total number of O(gn) moves is strictly smaller than that for the total gathering problem