Gathering an Even Number of Robots in an Odd Ring without Global Multiplicity Detection

Abstract

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...

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“…In [22], the case where k is odd and strictly smaller than n − 3 has been solved. In [23], the authors provide an algorithm for the case where n is odd, k is even, and 10 ≤ k ≤ n−5. Recently, the case of rigid configurations has been solved in [11].…”

On the self-stabilization of mobile oblivious robots in uniform rings

“…The algorithm assumes that initial configurations are non-symmetric and non-periodic, and the number of robots is less than half number of nodes. For odd number of robots or odd number of nodes in the same model, Kamei et al [14,15] propose the gathering algorithm that also works in symmetric configurations. Note that all of the above works assume some initial configurations and thus they are not self-stabilizing.…”

Universal Systems of Oblivious Mobile Robots

“…Indeed, since the network's port numbers may not be unique, it may be impossible for an algorithm to unambiguously indicate where each robot has to move. This model, introduced by Klasing, Markou, and Pelc [22] as an extension of the model of oblivious robots in continuous spaces (e.g., [14]), has been extensively employed and investigated, focusing on basic problems in specific classes of graphs under different schedulers: gathering and scattering (e.g., [4,5,6,7,10,16,17,19,21,22,25,26]), and exploration and traversal (e.g., [1,2,3,8,9,11,12,13,23,24]). Note that, with the exception of [3], the literature assumes unlabelled edges.…”