Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Advertisement

Springer Nature Link
Log in

Embedded pattern formation by asynchronous robots without chirality

  • Published:
Distributed Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

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 \(\ell _2\).

References

  1. Agmon, N., Peleg, D.: Fault-tolerant gathering algorithms for autonomous mobile robots. SIAM J. Comput. 36(1), 56–82 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bérard, B., Lafourcade, P., Millet, L., Potop-Butucaru, M., Thierry-Mieg, Y., Tixeuil, S.: Formal verification of mobile robot protocols. Distrib. Comput. 29(6), 459–487 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bhagat, S., Chaudhuri, S.G., Mukhopadhyaya, K.: Formation of general position by asynchronous mobile robots under one-axis agreement. In: Proceeding of 10th International ’l WS on Algorithms and Computation (WALCOM), LNCS, vol. 9627, pp. 80–91. Springer (2016)

  4. Bramas, Q., Tixeuil, S.: Brief Announcement: Probabilistic asynchronous arbitrary pattern formation. In: Proceedings of 35th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC) (2016)

  5. Cicerone, S., Di Stefano, G., Navarra, A.: Asynchronous embedded pattern formation without orientation. In: Proceedings of 30th International ’l Symposium on Distributed Computing (DISC), LNCS, vol. 9888, pp. 85–98. Springer (2016)

  6. Cicerone, S., Di Stefano, G., Navarra, A.: Asynchronous arbitrary pattern formation: the effects of a rigorous approach. Distrib. Comput. (2018). https://doi.org/10.1007/s00446-018-0325-7

    Article  MATH  Google Scholar 

  7. Cicerone, S., Di Stefano, G., Navarra, A.: Gathering of robots on meeting-points: feasibility and optimal resolution algorithms. Distrib. Comput. 31(1), 1–50 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by mobile robots: gathering. SIAM J. Comput. 41(4), 829–879 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Impossibility of gathering, a certification. Inf. Process. Lett. 115(3), 447–452 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Courtieu, P., Rieg, L., Tixeuil, S., Urbain, X.: Brief Announcement: Certified universal gathering in \(R^2\) for oblivious mobile robots. In: Proceedings of 35th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC) (2016)

  11. D’Angelo, G., Di Stefano, G., Klasing, R., Navarra, A.: Gathering of robots on anonymous grids and trees without multiplicity detection. Theor. Comput. Sci. 610, 158–168 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. D’Angelo, G., Di Stefano, G., Navarra, A.: Gathering on rings under the look-compute-move model. Distrib. Comput. 27(4), 255–285 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. D’Angelo, G., Di Stefano, G., Navarra, A., Nisse, N., Suchan, K.: Computing on rings by oblivious robots: a unified approach for different tasks. Algorithmica 72(4), 1055–1096 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. D’Angelo, G., Navarra, A., Nisse, N.: A unified approach for gathering and exclusive searching on rings under weak assumptions. Distrib. Comput. 30(1), 17–48 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Das, S., Flocchini, P., Santoro, N., Yamashita, M.: Forming sequences of geometric patterns with oblivious mobile robots. Distrib. Comput. 28(2), 131–145 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. D’Emidio, M., Frigioni, D., Navarra, A.: Characterizing the computational power of anonymous mobile robots. In: Proceedings of 36th IEEE International ’l Conference on Distributed Computing Systems, (ICDCS), pp. 293–302. IEEE (2016)

  17. Di Stefano, G., Navarra, A.: Gathering of oblivious robots on infinite grids with minimum traveled distance. Inf. Comput. 254, 377–391 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Di Stefano, G., Navarra, A.: Optimal gathering of oblivious robots in anonymous graphs and its application on trees and rings. Distrib. Comput. 30(2), 75–86 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dieudonné, Y., Labbani-Igbida, O., Petit, F.: Circle formation of weak mobile robots. TAAS 3(4), 16:1–16:20 (2008)

    Article  Google Scholar 

  20. Doan, H.T.T., Bonnet, F., Ogata, K.: Model checking of a mobile robots perpetual exploration algorithm. In: Proceedings of 6th International ’l Work on Structured Object-Oriented Formal Language and Method (SOFL+MSVL), Lecture Notes in Computer Science, vol. 10189, pp. 201–219 (2017)

  21. Donald, B.R., Jennings, J.S., Rus, D.: Information invariants for distributed manipulation. I. J. Robotics Res. 16(5), 673–702 (1997)

    Article  Google Scholar 

  22. Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, (2012)

  23. Flocchini, P., Prencipe, G., Santoro, N., Viglietta, G.: Distributed computing by mobile robots: uniform circle formation. Distrib. Comput. 30, 413–457 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci. 337, 147–168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Arbitrary pattern formation by asynchronous, anonymous, oblivious robots. Theor. Comput. Sci. 407(1–3), 412–447 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fujinaga, N., Ono, H., Kijima, S., Yamashita, M.: Pattern formation through optimum matching by oblivious corda robots. In: Proceedings of 14th International ’l Conference on Principles of Distributed Systems (OPODIS), LNCS, vol. 6490, pp. 1–15. Springer (2010)

  27. Fujinaga, N., Yamauchi, Y., Kijima, S., Yamashita, M.: Asynchronous pattern formation by anonymous oblivious mobile robots. In: Proceedings of 26th International ’l Symposium on Distributed Computing (DISC), LNCS, vol. 7611, pp. 312–325. Springer (2012)

  28. Fujinaga, N., Yamauchi, Y., Ono, H., Kijima, S., Yamashita, M.: Pattern formation by oblivious asynchronous mobile robots. SIAM J. Comput. 44(3), 740–785 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ghike, S., Mukhopadhyaya, K.: A distributed algorithm for pattern formation by autonomous robots, with no agreement on coordinate compass. In: Proceedings of 6th International ’l Conference on Distributed Computing and Internet Technology, (ICDCIT), LNCS, vol. 5966, pp. 157–169. Springer (2010)

  30. Izumi, T., Katayama, Y., Inuzuka, N., Wada, K.: Gathering autonomous mobile robots with dynamic compasses: An optimal result. In: Proceedings of 21st International ’l Symposium on Distributed Computing (DISC), LNCS, vol. 4731, pp. 298–312. Springer (2007)

  31. Klasing, R., Kosowski, A., Navarra, A.: Taking advantage of symmetries: gathering of many asynchronous oblivious robots on a ring. Theor. Comput. Sci. 411, 3235–3246 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mamino, M., Viglietta, G.: Square formation by asynchronous oblivious robots. In: Proceedings of the 28th Canadian Conference on Computational Geometry (CCCG), pp. 1–6 (2016)

  33. Millet, L., Potop-Butucaru, M., Sznajder, N., Tixeuil, S.: On the synthesis of mobile robots algorithms: The case of ring gathering. In: Proceedings of 16th International ’l Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), LNCS, vol. 8756, pp. 237–251. Springer (2014)

  34. Parker, L.E.: On the design of behavior-based multi-robot teams. Adv. Robot. 10(6), 547–578 (1995)

    Article  Google Scholar 

  35. Prencipe, G.: The effect of synchronicity on the behavior of autonomous mobile robot. Theory Comput. Syst. 38, 539–558 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Romanishin, J., Gilpin, K., Rus, D.: M-blocks: Momentum-driven, magnetic modular robots. In: Proceedings of IEEE/RSJ International ’l Conference on Intelligent Robots and Systems, pp. 4288–4295. IEEE (2013)

  37. Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28(4), 1347–1363 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yamashita, M., Suzuki, I.: Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theor. Comput. Sci. 411(26–28), 2433–2453 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yamauchi, Y., Uehara, T., Kijima, S., Yamashita, M.: Plane formation by synchronous mobile robots in the three dimensional euclidean space. In: Proceedings of 29th International ’l Symposium on Distributed Computing (DISC), LNCS, vol. 9363, pp. 92–106. Springer (2015)

  40. Yamauchi, Y., Uehara, T., Yamashita, M.: Brief Announcement: Pattern formation problem for synchronous mobile robots in the three dimensional euclidean space. In: Proceedings of 35th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC) (2016)

  41. Yamauchi, Y., Yamashita, M.: Randomized pattern formation algorithm for asynchronous oblivious mobile robots. In: Proceedings of 28th International ’l Symposium on Distributed Computing, (DISC), LNCS, vol. 8784, pp. 137–151. Springer (2014)

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriele Di Stefano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alfredo Navarra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

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”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-018-0333-7

Keywords

Search

Navigation