Abstract
The gathering problem has been largely studied in the last years with respect to different environments. The requirement is to move a team of robots initially placed at different locations toward a common point. Robots move based on the so called Look-Compute-Move model. Each time a robot wakes up, it perceives the current configuration in terms of robots’ positions (Look), it decides whether and where to move (Compute), and makes the computed move (Move) in the case that the decision was affirmative. All the phases are performed asynchronously for each robot. Robots are oblivious, anonymous, silent, and execute the same distributed and deterministic algorithm. So far, the goal has been mainly to detect the minimal assumptions that allow to accomplish the gathering task, without taking care of any cost measure of the provided solutions. We provide an overview of recent results that first extend the classic notion of optimization problem to the context of robot-based computing systems, and then show that the gathering problem can be optimally solved. As cost measure, the overall traveled distance performed by all robots is considered. This implies that the provided optimal algorithms must be able to solve the gathering by moving robots through shortest paths. The presented optimal algorithms refer to robots moving on either the plane or graphs. In the latter case, different topologies are considered, like trees, rings, and infinite grids.
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Notes
- 1.
In fact, asynchrony implies that, based on the configuration perceived during the Look phase at some time t, a robot r computes a destination at some time \(t' > t\), starts to move at an even later time \(t'' > t'\), eventually stopping at time \(t'''\ge t''\); thus it might be possible that at time \(t''\) some robots are in different positions from those previously perceived by r at time t, because in the meantime they performed their Move operations (possibly several times).
- 2.
More precisely, the adversary has also the power to stop a moving robot before it reaches its destination, but there exists an (unknown arbitrarily small) constant \(\delta > 0\) such that if the destination point is closer than \(\delta \), the robot will reach it, otherwise the robot will be closer to it by at least \(\delta \). Note that, without this assumption, an adversary would make it impossible for any robot to ever reach its destination.
- 3.
Meeting-points for gathering purposes are interesting not only from a theoretical point of view, but also for practical reasons when not all places can be candidate to serve as gathering points.
- 4.
The subscript in the symbol \(V_r^+(p)\) is used to remark who is computing the view (in this case r), while the argument indicates the point from which the view is computed.
- 5.
If two points \(r'\in U(R)\) and \(m\in M\), different from p, are coincident, then points \(r',m\) will appear in this order in \(V_r^+(p) \).
- 6.
Remember that the terms clockwise and counter-clockwise always refer to the local coordinate system of the robot that computes the view. During a computational cycle, r maintains the same local orientation to compute the view of each point \(p\in R\cup M\), but the orientation could change between two different computational cycles.
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Acknowledgements
The work has been supported in part by the European project “Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies” (GEO-SAFE), contract no. H2020-691161 and by the Italian project “RISE: un nuovo framework distribuito per data collection, monitoraggio e comunicazioni in contesti di emergency response”, Fondazione Cassa Risparmio Perugia, code 2016.0104.021.
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Cicerone, S., Di Stefano, G., Navarra, A. (2018). Gathering a Swarm of Robots Through Shortest Paths. In: Adamatzky, A. (eds) Shortest Path Solvers. From Software to Wetware. Emergence, Complexity and Computation, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-319-77510-4_2
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