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Gathering on rings under the Look–Compute–Move model

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Abstract

A set of robots arbitrarily placed on different nodes of an anonymous ring have to meet at one common node and there remain. This problem is known in the literature as the gathering. Anonymous and oblivious robots operate in Look–Compute–Move cycles; in one cycle, a robot takes a snapshot of the current configuration (Look), decides whether to stay idle or to move to one of its neighbors (Compute), and in the latter case makes the computed move instantaneously (Move). Cycles are asynchronous among robots. Moreover, each robot is empowered by the so called multiplicity detection capability, that is, it is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one. The described problem has been extensively studied during the last years. However, the known solutions work only for specific initial configurations and leave some open cases. In this paper, we provide an algorithm which solves the general problem but for few marginal and specific cases, and is able to detect all the ungatherable configurations. It is worth noting that our new algorithm makes use of some previous techniques and unifies them with new strategies in order to deal with any initial configuration, even those left open by previous works.

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Notes

  1. Such an odd interval always exists. In fact, if the axis of symmetry passes through two even intervals the configuration has an edge–edge symmetry which is ungatherable.

  2. Actually the graphical descriptions contained in “Appendix 1” can be exploited by the reader for all configuration types and hence from now on we do not point again to such appendix.

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Authors and Affiliations

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

    Gianlorenzo D’Angelo & Alfredo Navarra

  2. Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100 , L’Aquila, Italy

    Gianlorenzo D’Angelo

  3. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via G. Gronchi 18, L’Aquila, 67100, Italy

    Gabriele Di Stefano

Authors
  1. Gianlorenzo D’Angelo
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  2. Gabriele Di Stefano
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  3. Alfredo Navarra
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Correspondence to Gianlorenzo D’Angelo.

Additional information

Work supported by the Research Grant 2010N5K7EB ‘PRIN 2010’ ARS TechnoMedia (Algoritmica per le Reti Sociali Tecno-mediate) from the Italian Ministry of University and Research. Preliminary results concerning this work have been presented in [10].

Appendices

Appendix 1: Transitions among types of configuration

In this appendix we provide a graphical representation of the possible transitions among all the types of configurations. In particular, for each configuration type we show all the configurations that can be reached according to the algorithm and the asynchronous execution of the Look–Compute–Move cycles.

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Appendix 2: Special cases for configuration transitions

In this section, we describe the behavior of the algorithm in the cases that lead to backward arcs in Fig. 2.

1.1 From multiplicity–convergence to multiplicity–creation

The only case when a configuration in multiplicity–convergence can lead to one of multiplicity–creation is that with six robots, that is the initial configuration was in W3. An exhaustive example is given below.

Let us consider the configuration in Mc-s-x a given in Fig. 21a where each multiplicity contains two robots (and hence there are six robots in the ring). The algorithm aims to move the two multiplicity towards the north. However, it may happen that only one robot moves from each multiplicity, hence obtaining the configuration in W3 given in Fig. 21b. At this point, the algorithm in [9] is applied which leads again to the configuration in Mc-s-x a given in Fig. 21d, possibly passing through that in Fig. 21c which belongs to Mc-a-1. Therefore, in these cases, this process can be repeated a finite number of times, until the two multiplicities join into the north, hence the backward arc from multiplicity–convergence to multiplicity–creation of Fig. 2 can be traversed a finite number of times.

Fig. 21
figure 21

Configurations on type: (a) Mc-s-x a, (b) W3, (c) Mc-a-1, (d) Mc-s-x a

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Fig. 22
figure 22

Configurations on type: (a) Conv-a-1 b, (b) Coll-a-1, (c) Coll-s-2 c, (d) Mc-s-x a, (e) Conv-s-1

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1.2 From convergence to collect

The only case when a configuration in convergence can lead to one of collect is that with more than six nodes occupied where an xn move leads to a configuration at one reduction move from a symmetric configuration. That is we can go from a configuration in Conv-a-1 b to one in Coll-a-1. An exhaustive example is given below.

Let us consider the configuration in Conv-a-1 b given in Fig. 21a. In this case, the algorithm performs an xn move, leading to the configuration given in Fig. 22b. Note that such a configuration belongs to Coll-a-1 as it is at one reduction move from the symmetric configuration in Coll-s-2 c given in Fig. 22c. Therefore, the algorithm forces such a reduction move, obtaining the configuration in Fig. 22c. Then, the two single robots which are not guards are moved to join the multiplicities. At this point (see Fig. 22d) each multiplicity contains at least three robots and therefore both of them are moved towards the north in phase multiplicity–convergence. Since each multiplicity contains at least three robots, this phase cannot generate configurations with only one multiplicity, except for the last steps when the two multiplicities are joint (see e.g. Fig. 22e). This implies that moving from convergence to collect can occur only once and therefore the backward arc from convergence to collect of Fig. 2 can be traversed only once.

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D’Angelo, G., Di Stefano, G. & Navarra, A. Gathering on rings under the Look–Compute–Move model. Distrib. Comput. 27, 255–285 (2014). https://doi.org/10.1007/s00446-014-0212-9

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