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Gathering of robots on meeting-points: feasibility and optimal resolution algorithms

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Abstract

The paper considers variants of the gathering problem of oblivious and asynchronous robots moving in the plane. When \(n>2\) robots are free to gather anywhere in the plane, the problem has been solved in Cieliebak et al. (SIAM J Comput 41(4):829–879, 2012). We propose a new natural and challenging model that requires robots to gather only at some predetermined points in the plane, referred to as meeting-points. Robots operate in standard Look-Compute-Move cycles. In one cycle, a robot perceives the current configuration in terms of robots’ positions and meeting-points (Look) according to its own coordinate system, decides whether to move toward some direction (Compute), and in the positive case it moves (Move). Cycles are performed asynchronously for each robot. Robots are anonymous and execute the same distributed and deterministic algorithm. In the new proposed model, we fully characterize when gathering can be accomplished. We design an algorithm that solves the problem for all configurations with \(n>0\) robots but those identified as ungatherable. After that, we consider the classical notion of optimization algorithms and extend it to the context of robot-based computing systems. With this new notion, we re-consider the gathering on meeting-points problem but with respect to two objective functions. In particular, we first solve the gathering by minimizing the overall traveled distance performed by all robots and then we address the minimization of the maximum traveled distance performed by a single robot. For the former objective function, we fully characterize when optimal gathering can be achieved by providing a distributed algorithm along with the proof of correctness. For the latter objective function, we design another gathering algorithm that ensures optimal gathering almost for all the cases where it is possible, and discuss some insights on the remaining cases.

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Notes

  1. 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.

  2. 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 \(V_r^+(p) \).

  3. 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.

  4. Configurations in class \(\mathscr {S}^{+}_5\), that is all robots and all Weber-points are collinear, have been already addressed.

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Authors and Affiliations

  1. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, Coppito, 67100, L’Aquila, Italy

    Serafino Cicerone & Gabriele Di Stefano

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Alfredo Navarra

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  1. Serafino Cicerone
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  2. Gabriele Di Stefano
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  3. Alfredo Navarra
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Correspondence to Serafino Cicerone.

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Preliminary versions of this paper appeared in Proceedings of the 10th and 11th International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (Algosensors ’14 and ’15) [21, 22], and in Proceedings of the 9th International Conference on Algorithms and Complexity (CIAC’15) [23].

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, by the Italian Ministry of Education, University, and Research (MIUR), by the project PRIN 2012C4E3KT “AMANDA—Algorithmics for MAssive and Networked DAta”, 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. Gathering of robots on meeting-points: feasibility and optimal resolution algorithms. Distrib. Comput. 31, 1–50 (2018). https://doi.org/10.1007/s00446-017-0293-3

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