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A unified approach for gathering and exclusive searching on rings under weak assumptions

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Abstract

Consider a set of mobile robots placed on distinct nodes of a discrete, anonymous, and bidirectional ring. Asynchronously, each robot takes a snapshot of the ring, determining the size of the ring and which nodes are either occupied by robots or empty. Based on the observed configuration, it decides whether to move to one of its adjacent nodes or not. In the first case, it performs the computed move, eventually. This model of computation is known as Look-Compute-Move. The computation depends on the required task. In this paper, we solve both the well-known Gathering and Exclusive Searching tasks. In the former problem, all robots must simultaneously occupy the same node, eventually. In the latter problem, the aim is to clear all edges of the graph. An edge is cleared if it is traversed by a robot or if both its endpoints are occupied. We consider the exclusive searching where it must be ensured that two robots never occupy the same node. Moreover, since the robots are oblivious, the clearing is perpetual, i.e., the ring is cleared infinitely often. In the literature, most contributions are restricted to a subset of initial configurations. Here, we design two different algorithms and provide a characterization of the initial configurations that permit the resolution of the problems under very weak assumptions. More precisely, we provide a full characterization (except for few pathological cases) of the initial configurations for which gathering can be solved. The algorithm relies on the necessary assumption of the local-weak multiplicity detection. This means that during the Look phase a robot detects also whether the node it occupies is occupied by other robots, without acquiring the exact number. For the exclusive searching, we characterize all (except for few pathological cases) aperiodic configurations from which the problem is feasible. We also provide some impossibility results for the case of periodic configurations.

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Notes

  1. Configuration \({{\mathcal {C}}} = (0,1,0,1,1,0,1,1)\) is the only exception, see Sect. 6.3.

  2. Except for the case of \({{\mathcal {C}}}^{\min } = (0^{i},1,0^x,1,(0^{i},1,0^{x-1},1)^{\ell })\) for \(\ell =1\) and \(x=1\) (i.e. \({{\mathcal {C}}}^{\min } = (0^i,1,0,1,0^i,1,1)\)) where \({\textsc {reduce}} _1\) is performed. This case will be discussed in Sect. 6.3, along with the case \({{\mathcal {C}}}^{\min } = (0^{i},1,0^x,1,0^{i},1,0^{y},1)\), \(y<x\).

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Authors and Affiliations

  1. Gran Sasso Science Institute (GSSI), Via F. Crispi, 7, 67100, L’Aquila, Italy

    Gianlorenzo D’Angelo

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

    Alfredo Navarra

  3. INRIA/I3S(CNRS/UNSA), 2004, route des Lucioles, B.P. 93, 06902, Sophia Antipolis Cedex, France

    Nicolas Nisse

Authors
  1. Gianlorenzo D’Angelo
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  2. Alfredo Navarra
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  3. Nicolas Nisse
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Correspondence to Gianlorenzo D’Angelo.

Additional information

Preliminary results concerning this work were presented in the 15th International Conference on Distributed Computing and Networking (ICDCN’14) [15].

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D’Angelo, G., Navarra, A. & Nisse, N. A unified approach for gathering and exclusive searching on rings under weak assumptions. Distrib. Comput. 30, 17–48 (2017). https://doi.org/10.1007/s00446-016-0274-y

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