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 ring-shaped 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(n 2 ), 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).

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A loosely-stabilizing leader election protocol with polylogarithmic convergence time in the population protocol model is presented in this paper. In the population protocol model, which is a common abstract model of mobile sensor networks, it is known to be impossible to design a self-stabilizing leader election protocol. Thus, in our prior work, we introduced the concept of loose-stabilization, which is weaker than self-stabilization but has similar advantage as selfstabilization in practice. Following this work, several loosely-stabilizing leader election protocols are presented. The loosely-stabilizing leader election guarantees that, starting from an arbitrary configuration, the system reaches a safe configuration with a single leader within a relatively short time, and keeps the unique leader for an sufficiently long time thereafter. The convergence times of all the existing loosely-stabilizing protocols, i.e., the expected time to reach a safe configuration, are polynomial in n where n is the number of nodes (while the holding times to keep the unique leader are exponential in n). In this paper, a loosely-stabilizing protocol with polylogarithmic convergence time is presented. Its holding time is not exponential, but arbitrarily large polynomial in n.

In this article, we present the first leader election protocol in the population protocol model that stabilizes within Oðlog nÞ parallel time in expectation with Oðlog nÞ states per agent, where n is the number of agents. Given a rough knowledge m of lg n such that m ! lg n and m ¼ Oðlog nÞ, the proposed protocol guarantees that exactly one leader is elected and the unique leader is kept forever thereafter. This protocol is time-optimal because it was recently proven that any leader election protocol requires Vðlog nÞ parallel time.

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In this paper, we present the first leader election protocol in the population protocol model that stabilizes O(log n) parallel time in expectation with O(log n) states per agent, where n is the number of agents. Given a rough knowledge m of the population size n such that m ≥ log 2 n and m = O(log n), the proposed protocol guarantees that exactly one leader is elected and the unique leader is kept forever thereafter.

We propose a gathering protocol for an even 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 nonatomic CORDA model with asynchronous fair scheduling. In our scheme, the number of robots k must be greater than 8, the number of nodes n on a network must be odd and greater than k + 3. The running time of our protocol is O(n 2 ) asynchronous rounds. This work is supported in part by KAKENHI no.22700074. so (i.e., the two locations are connected by an edge in the representing graph). The discrete model permits to simplify many robot protocols by reasoning on finite structures (i.e., graphs) rather than on infinite ones. Related Work. In this paper, we focus on the gathering problem in the discrete setting where a set of robots has to gather in one single location, not defined in advance, and remain on this location [4,2,6,7,5,1]. Several deterministic algorithms have been proposed to solve the gathering problem in a ring-shaped network, which enables many problems to appear due to the high number of symmetric configurations. The case of anonymous, asynchronous and oblivious robots was investigated only recently in this context. It should be noted that if the configuration is periodic and edge symmetric, no deterministic solution can exist [7]. The first two solutions [7,6] are complementary: [7] is based on breaking the symmetry whereas [6] takes advantage of symmetries. However, both [7] and [6] make the assumption that robots are endowed with the ability to distinguish nodes that host one robot from nodes that host two robots or more in the entire network (this property is referred to in the literature as global weak multiplicity detection). This ability weakens the gathering problem because it is sufficient for a protocol to ensure that a single multiplicity point exists to have all robots gather in this point, so it reduces the gathering problem to the creation of a single multiplicity point. Nevertheless, the case of an even number of robots proved difficult [3, 1] as more symmetric situations must be taken into account.Investigating the feasibility of gathering with weaker multiplicity detectors was recently addressed in [4,5]. In those papers, robots are only able to test that their current hosting node is a multiplicity node (i.e. hosts at least two robots). This assumption (referred to in the literature as local weak multiplicity detection) is obviously weaker than the global weak multiplicity detection, but is also more realistic as far as sensing devices are concerned. The downside of [4] compared to [6] is that only rigid configurations (i.e. non symmetric configuration) are allowed as initial configurations (as in [7]), while [6] allowed symmetric but not periodic configurations to be used as initial ones. Also, [4] requires that k < n/2 even in the case of non-symmetric configurations, where k denotes th...