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Embedded pattern formation by asynchronous robots without chirality

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Abstract

We consider the Embedded Pattern Formation (epf) problem introduced in Fujinaga et al. (SIAM J Comput 44(3):740–785, 2015). Given a set F of distinct points in the Euclidean plane (called here fixed-points) and a set R of robots such that \(|R|=|F|\), the problem asks for a distributed algorithm that moves robots so as to occupy all points in F. Initially, each robot occupies a distinct position. When active, a robot operates in standard Look-Compute-Move cycles. In one cycle, a robot perceives the current configuration in terms of the robots’ positions and the fixed-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 oblivious, anonymous, silent and execute the same deterministic algorithm. In the mentioned paper, the problem has been investigated by endowing robots with chirality, that is they share a common left-right orientation. Here we consider epf without chirality, and we fully characterize when it can be solved by designing a deterministic distributed algorithm that works for all configurations but those identified as unsolvable. The algorithm has been designed according to a rigorous approach, characterized by the use of logical predicates associated to each move used by the robots. This induces a greater level of detail that provides us rigorous bases to state the correctness of the algorithm.

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Notes

  1. Note that in [37, 38], it is assumed chirality and that the direction of the positive x-axis and the magnitude of the unit distance never change, but differently from [28] each robot refers to an LCS not necessarily centered in its current location.

  2. We can assume \(t_i = i\) for all \(i = 0,1,\ldots \) since the information relevant in the definition of execution is the order in which the different snapshots occur and not the exact time in which each snapshots is taken.

  3. Here the term ‘phase’ is informally used to denote a generic part of an algorithm and hence it is not referred to the definition of phase provided in Sect. 2.

  4. By Corollary 2, axes with only fixed-points cannot exist as the initial configuration was solvable and the moves performed from the initial configuration until the current step guarantee to not create such kind of axes.

  5. \(\sphericalangle (\ell _1,\ell _2)\) is the smallest angle defined by the half-lines \(\ell _1\) and \(\ell _2\).

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Authors and Affiliations

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

    Serafino Cicerone & Gabriele Di Stefano

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

    Alfredo Navarra

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

Additional information

Preliminary results appeared in the Proceedings of the 30th Int.’l Symp. on Distributed Computing (DISC) 2016 [5]. 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 National Group for Scientific Computation (GNCS-INdAM) research project 2018 “Anti-Social Networks”.

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Cicerone, S., Di Stefano, G. & Navarra, A. Embedded pattern formation by asynchronous robots without chirality. Distrib. Comput. 32, 291–315 (2019). https://doi.org/10.1007/s00446-018-0333-7

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