Asynchronous Mobile Robot Gathering from Symmetric Configurations without Global Multiplicity Detection

We consider a set of k autonomous robots that are endowed with visibility sensors (but that are otherwise unable to communicate) and motion actuators. Those robots must collaborate to reach a single vertex that is unknown beforehand, and to remain there hereafter. Previous works on gathering in ringshaped networks suggest that there exists a tradeoff between the size of the set of potential initial configurations, and the power of the sensing capabilities of the robots (i.e. the larger the initial configuration set, the most powerful the sensor needs to be). We prove that there is no such trade off. We propose a gathering protocol for an odd number of robots in a ring-shaped network that allows symmetric but not periodic configurations as initial configurations, yet uses only local weak multiplicity detection. Robots are assumed to be anonymous and oblivious, and the execution model is the non-atomic CORDA model with asynchronous fair scheduling. Our protocol allows the largest set of initial configurations (with respect to impossibility results) yet uses the weakest multiplicity detector to date. The time complexity of our protocol is O(n2), where n denotes the size of the ring. Compared to previous work that also uses local weak multiplicity detection, we do not have the constraint that k < n/2 (here, we simply have 2 < k < n-3).