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Asynchronous Arbitrary Pattern Formation: the effects of a rigorous approach

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Abstract

Given any multiset F of points in the Euclidean plane and a set R of robots such that \(|R|=|F|\), the Arbitrary Pattern Formation (APF) problem asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections, uniform scalings. Initially, each robot occupies a distinct position. When active, a robot operates in standard Look–Compute–Move cycles. Robots are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. So far, the problem has been mainly addressed by assuming chirality, that is robots share a common left–right orientation. We are interested in removing such a restriction. While working on the subject, we faced several issues that required close attention. We deeply investigated how such difficulties were overcome in the literature, revealing that crucial arguments for the correctness proof of the existing algorithms have been neglected. The systematic lack of rigorous arguments with respect to necessary conditions required for providing correctness proofs deeply affects the validity as well as the relevance of strategies proposed in the literature. Here we design a new deterministic distributed algorithm that fully characterizes APF showing its equivalence with the well-known Leader Election problem in the asynchronous model without chirality. Our approach is characterized by the use of logical predicates in order to formally describe our algorithm as well as its correctness. In addition to the relevance of our achievements, our techniques might help in revising previous results. In fact, it comes out that well-established results like (Fujinaga et al. in SIAM J Comput 44(3):740–785, 2015), more recent approaches like (Bramas and Tixeuil, in: Proceedings of the 35th ACM SIGACT-SIGOPS symposium on principles of distributed computing (PODC), 2016; Bramas and Tixeuil, in: Proceedings of the 18th international symposium on stabilization, safety, and security of distributed systems (SSS), 2016) and ‘unofficial’ results like (Dieudonné et al., in: CoRR arXiv:0902.2851, 2009) revealed to be not correct.

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Notes

  1. The definition of stationary robot provided in [22] is slightly different but also inaccurate. In fact, it does not catch the third scenario about active robots described by our definition. If removing such a case, no configuration might be declared stationary during an execution. Similarly, the definition of static robot from [5] does not describe for instance the case where a robot is not moving but has already performed the Look phase.

  2. Note that in this work we use operations on multisets.

  3. When chirality can be exploited, reflections can be ignored as \(V^+(p)\) can be always discriminated from \(V^-(q)\).

  4. Indeed, this is not necessary when chirality is assumed.

  5. In the literature, this is sometime realized by inducing a common coordinate system. This method can be effective only if F is specified by coordinates and not by distances.

  6. The technique adopted by means of Procedure StationaryMove is similar to the so-called cautious move in [20, 25].

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Authors and Affiliations

  1. Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via Vetoio, 67100, L’Aquila, Italy

    Serafino Cicerone & Gabriele Di Stefano

  2. Department of Mathematics and Computer Science, University of 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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Corresponding author

Correspondence to Alfredo Navarra.

Additional information

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).

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Cicerone, S., Di Stefano, G. & Navarra, A. Asynchronous Arbitrary Pattern Formation: the effects of a rigorous approach. Distrib. Comput. 32, 91–132 (2019). https://doi.org/10.1007/s00446-018-0325-7

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  • DOI: https://doi.org/10.1007/s00446-018-0325-7

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