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Shape Formation by Programmable Particles

Authors Giuseppe A. Di Luna, Paola Flocchini, Nicola Santoro, Giovanni Viglietta, Yukiko Yamauchi

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Giuseppe A. Di Luna
Paola Flocchini
Nicola Santoro
Giovanni Viglietta
Yukiko Yamauchi

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Giuseppe A. Di Luna, Paola Flocchini, Nicola Santoro, Giovanni Viglietta, and Yukiko Yamauchi. Shape Formation by Programmable Particles. In 21st International Conference on Principles of Distributed Systems (OPODIS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 95, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.OPODIS.2017.31

Abstract

Shape formation (or pattern formation) is a basic distributed problem for systems of compu- tational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter, where entities are assumed to be small and with severely limited capabilities. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessella- tion of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, start- ing from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization (i.e., particles can flip coins to elect a leader).
In this paper we provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of n particles. The characterization is constructive: we provide a universal shape formation algorithm that, for each feasible pair of shapes (S_0,S_F), allows the particles to form the final shape SF (given in input) starting from the initial shape S_0, unknown to the particles. The final configuration will be an appropriate scaled-up copy of S_F depending on n.
If randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that there are enough particles.
Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential.
We also consider the complexity of shape formation both in terms of the number of rounds and the total number of moves performed by the particles executing a universal shape formation algorithm. We prove that our solution has a complexity of O(n^2) rounds and moves: this number of moves is also asymptotically worst-case optimal.

Subject Classification

Keywords
  • Shape formation
  • pattern formation
  • programmable matter
  • Amoebots
  • leader election
  • distributed algorithms

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References

  1. H. Ando, I. Suzuki, and M. Yamashita. Formation and agreement problems for synchronous mobile robots with limited visibility. In Proc. of ISIC, pages 453-460, 1995. Google Scholar
  2. S. Cannon, J.J. Daymude, D. Randall, and A.W. Richa. A Markov chain algorithm for compression in self-organizing particle systems. In Proc. of PODC, pages 279-288, 2016. Google Scholar
  3. G. Chirikjian. Kinematics of a metamorphic robotic system. In Proc. of ICRA, pages 1:449-1:455, 1994. Google Scholar
  4. S. Das, P. Flocchini, N. Santoro, and M. Yamashita. Forming sequences of geometric patterns with oblivious mobile robots. Distrib. Comput., 28(2):131-145, 2015. Google Scholar
  5. J.J. Daymude, Z. Derakhshandeh, R. Gmyr, A. Porter, A.W. Richa, C. Scheideler, and T. Strothmann. On the runtime of universal coating for programmable matter. In Proc. of DNA, pages 148-164, 2016. Google Scholar
  6. J.J. Daymude, R. Gmyr, A.W. Richa, C. Scheideler, and T. Strothmann. Improved leader election for self-organizing programmable matter. arXiv, 2017. URL: https://arxiv.org/abs/1701.03616.
  7. Z. Derakhshandeh, R. Gmyr, A.W. Richa, C. Scheideler, and T. Strothmann. An algorithmic framework for shape formation problems in self-organizing particle systems. In Proc. of NanoCom, pages 21:1-21:2, 2015. Google Scholar
  8. Z. Derakhshandeh, R. Gmyr, A.W. Richa, C. Scheideler, and T. Strothmann. Universal shape formation for programmable matter. In Proc. of SPAA, pages 289-299, 2016. Google Scholar
  9. Z. Derakhshandeh, R. Gmyr, A.W. Richa, C. Scheideler, and T. Strothmann. Universal coating for programmable matter. Theor. Comput. Sci., 671:56-68, 2017. Google Scholar
  10. Z. Derakhshandeh, R. Gmyr, T. Strothmann, R.A. Bazzi, A.W. Richa, and C. Scheideler. Leader election and shape formation with self-organizing programmable matter. In Proc. of DNA, pages 117-132, 2015. Google Scholar
  11. S. Dolev, S. Frenkel, M. Rosenbli, P. Narayanan, and K.M. Venkateswarlu. In-vivo energy harvesting nano robots. In Proc. of ICSEE, pages 1-5, 2016. Google Scholar
  12. P. Flocchini, G. Prencipe, N. Santoro, and P. Widmayer. Arbitrary pattern formation by asynchronous, anonymous, oblivious robots. Theor. Comput. Sci., 407(1):412-447, 2008. Google Scholar
  13. N. Fujinaga, Y. Yamauchi, H. Ono, S. Kijima, and M. Yamashita. Pattern formation by oblivious asynchronous mobile robots. SIAM J. Comput., 44(3):740-785, 2016. Google Scholar
  14. G.A. Di Luna, P. Flocchini, G. Prencipe, N. Santoro, and G. Viglietta. Line recovery by programmable particles. In Proc. of ICDCN, to appear. Google Scholar
  15. G.A. Di Luna, P. Flocchini, N. Santoro, G. Viglietta, and Y. Yamauchi. Shape formation by programmable particles. arXiv, 2017. URL: https://arxiv.org/abs/1705.03538.
  16. O. Michail, G. Skretas, and P.G. Spirakis. On the transformation capability of feasible mechanisms for programmable matter. In Proc. of ICALP, pages 136:1-136:15, 2017. Google Scholar
  17. A. Naz, B. Piranda, J. Bourgeois, and S.C. Goldstein. A distributed self-reconfiguration algorithm for cylindrical lattice-based modular robots. In Proc. of NCA, pages 254-263, 2016. Google Scholar
  18. M.J. Patitz. An introduction to tile-based self-assembly and a survey of recent results. Nat. Comput., 13(2):195-224, 2014. Google Scholar
  19. P.W. Rothemund. Folding DNA to create nanoscale shapes and patterns. Nature, 440(7082):297-302, 2006. Google Scholar
  20. M. Rubenstein, A. Cornejo, and R. Nagpal. Programmable self-assembly in a thousand-robot swarm. Science, 345(6198):795-799, 2014. Google Scholar
  21. N. Schiefer and E. Winfree. Universal computation and optimal construction in the chemical reaction network-controlled tile assembly model. In Proc. of DNA, pages 34-54, 2015. Google Scholar
  22. I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: formation of geometric patterns. SIAM J. Comput., 28(4):1347-1363, 1999. Google Scholar
  23. T. Toffoli and N. Margolus. Programmable matter: concepts and realization. Physica D, 47(1):263-272, 1991. Google Scholar
  24. J.E. Walter, J.L. Welch, and N.M. Amato. Distributed reconfiguration of metamorphic robot chains. Distrib. Comput., 17(2):171-189, 2004. Google Scholar
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