Mobile Robots Gathering Algorithm with Local Weak Multiplicity in Rings

Gathering of Six Robots on Anonymous Symmetric Rings

“…Under the Look-Compute-Move model, the gathering problem on rings was initially studied in [7], where certain configurations were shown to be gatherable by means of symmetry-breaking techniques, but the question of the general-case solution was posed as an open problem. In particular, it has been proved that the gathering is not feasible in configurations with only two robots, in periodic configurations (invariable under non-trivial rotation) or in those with an axis of symmetry of type edge-edge.…”

“…A configuration is called symmetric if the ring has a geometrical axis of symmetry, which reflects single robots into single robots, multiplicities into multiplicities, and empty nodes into empty nodes. A symmetric configuration is not periodic if and only if it has exactly one axis of symmetry [7]. A symmetric configuration with an axis of symmetry has an edge-edge symmetry if the axis goes through (the middles of) two edges; it has a node-edge symmetry if the axis goes through one node and one edge; it has a node-node symmetry if the axis goes through two nodes; it has a robot-on-axis symmetry if there is at least one node on the axis of symmetry occupied by a robot.…”

“…The new proposed technique was based on preserving symmetry rather than breaking it, and the problem was solved when the number of robots is greater than 18. This left open the case of an even number of robots between 4 and 18, as the case of just 2 robots is not gatherable [7]. The case of 4 robots has been solved in [3,8].…”

“…Moreover, it is assumed that the robots have the ability to perceive whether there is one or more robots located at a given node of the ring. This capability of robots is important and well-studied in the literature under the name of multiplicity detection [3,6,7,2,4,5,9]. In fact, without this capability, many computational problems (such as the gathering problem considered herein) are impossible to solve for all non-trivial starting configurations.…”

“…In particular, we consider the case of only six robots placed on the nodes of an anonymous ring in such a way they constitute a symmetric placement with respect to one single axis of symmetry, and we ask whether there exists a strategy that allows the robots to gather at one single node. This is in fact the first case left open after a series of papers [3,[6][7][8] dealing with the gathering of oblivious robots on anonymous rings. As long as the gathering is feasible, we provide a new distributed approach that guarantees a positive answer to the posed question.…”