Skip to main content
SpringerLink
Log in

On Efficient Connectivity-Preserving Transformations in a Grid

  • Conference paper
  • First Online:
Algorithms for Sensor Systems (ALGOSENSORS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 12503))

Abstract

We consider a discrete system of n devices lying on a 2-dimensional square grid and forming an initial connected shape \(S_I\). Each device is equipped with a linear-strength mechanism which enables it to move a whole line of consecutive devices in a single time-step. We study the problem of transforming \(S_I\) into a given connected target shape \(S_F\) of the same number of devices, via a finite sequence of line moves. Our focus is on designing centralised transformations aiming at minimising the total number of moves subject to the constraint of preserving connectivity of the shape throughout the course of the transformation. We first give very fast connectivity-preserving transformations for the case in which the associated graphs of \( S_I \) and \( S_F \) are isomorphic to a Hamiltonian line. In particular, our transformations make \( O(n \log n \)) moves, which is asymptotically equal to the best known running time of connectivity-breaking transformations. Our most general result is then a connectivity-preserving universal transformation that can transform any initial connected shape \( S_I \) into any target connected shape \( S_F \), through a sequence of \(O(n\sqrt{n})\) moves.

The full version of the paper is available at: https://arxiv.org/abs/2005.08351.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akitaya, H., et al.: Universal reconfiguration of facet-connected modular robots by pivots: the O(1) musketeers. In: 27th Annual European Symposium on Algorithms, ESA. LIPIcs, vol. 144, pp. 3:1–3:14 (2019)

    Google Scholar 

  2. Almethen, A., Michail, O., Potapov, I.: On efficient connectivity-preserving transformations in a grid. CoRR abs/2005.08351 (2020)

    Google Scholar 

  3. Almethen, A., Michail, O., Potapov, I.: Pushing lines helps: efficient universal centralised transformations for programmable matter. Theoret. Comput. Sci. 830–831, 43–59 (2020)

    Article  MathSciNet  Google Scholar 

  4. Aloupis, G., et al.: Efficient reconfiguration of lattice-based modular robots. Comput. Geom. 46(8), 917–928 (2013)

    Article  MathSciNet  Google Scholar 

  5. Aloupis, G., Collette, S., Demaine, E.D., Langerman, S., Sacristán, V., Wuhrer, S.: Reconfiguration of cube-style modular robots using O(logn) parallel moves. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 342–353. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-92182-0_32

    Chapter  Google Scholar 

  6. Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18(4), 235–253 (2006). https://doi.org/10.1007/s00446-005-0138-3

    Article  MATH  Google Scholar 

  7. Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: The computational power of population protocols. Distrib. Comput. 20(4), 279–304 (2007). https://doi.org/10.1007/s00446-007-0040-2

    Article  MATH  Google Scholar 

  8. Arora, S., Raghavan, P., Rao, S.: Approximation schemes for Euclidean k-medians and related problems. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of computing, pp. 106–113 (1998)

    Google Scholar 

  9. Bourgeois, J., Goldstein, S.: Distributed intelligent MEMS: progresses and perspective. IEEE Syst. J. 9(3), 1057–1068 (2015)

    Article  Google Scholar 

  10. Butler, Z., Kotay, K., Rus, D., Tomita, K.: Generic decentralized control for lattice-based self-reconfigurable robots. Int. J. Robot. Res. 23(9), 919–937 (2004)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Clementi, A., Di Ianni, M., Lauria, M., Monti, A., Rossi, G., Silvestri, R.: On the bounded-hop MST problem on random Euclidean instances. Theoret. Comput. Sci. 384(2–3), 161–167 (2007)

    Article  MathSciNet  Google Scholar 

  13. Czyzowicz, J., Dereniowski, D., Pelc, A.: Building a nest by an automaton. In: Bender, M.A., Svensson, O., Herman, G. (eds.) 27th Annual European Symposium on Algorithms, ESA (2019)

    Google Scholar 

  14. Das, S., Flocchini, P., Santoro, N., Yamashita, M.: Forming sequences of geometric patterns with oblivious mobile robots. Distrib. Comput. 28(2), 131–145 (2014). https://doi.org/10.1007/s00446-014-0220-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Daymude, J.J., et al.: On the runtime of universal coating for programmable matter. Nat. Comput. 17(1), 81–96 (2017). https://doi.org/10.1007/s11047-017-9658-6

    Article  MathSciNet  Google Scholar 

  16. Derakhshandeh, Zahra., Gmyr, Robert., Porter, Alexandra., Richa, Andréa W., Scheideler, Christian, Strothmann, Thim: On the runtime of universal coating for programmable matter. In: Rondelez, Yannick, Woods, Damien (eds.) DNA 2016. LNCS, vol. 9818, pp. 148–164. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43994-5_10

    Chapter  Google Scholar 

  17. Derakhshandeh, Z., Gmyr, R., Richa, A., Scheideler, C., Strothmann, T.: Universal shape formation for programmable matter. In: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 289–299. ACM (2016)

    Google Scholar 

  18. Di Luna, G.A., Flocchini, P., Santoro, N., Viglietta, G., Yamauchi, Y.: Shape formation by programmable particles. Distrib. Comput. 33(1), 69–101 (2019). https://doi.org/10.1007/s00446-019-00350-6

    Article  MathSciNet  MATH  Google Scholar 

  19. Doty, D.: Theory of algorithmic self-assembly. Commun. ACM 55, 78–88 (2012)

    Article  Google Scholar 

  20. Douglas, S., Dietz, H., Liedl, T., Högberg, B., Graf, F., Shih, W.: Self-assembly of DNA into nanoscale three-dimensional shapes. Nature 459(7245), 414 (2009)

    Article  Google Scholar 

  21. Dumitrescu, A., Pach, J.: Pushing squares around. In: Proceedings of the Twentieth Annual Symposium on Computational Geometry, pp. 116–123. ACM (2004)

    Google Scholar 

  22. Dumitrescu, A., Suzuki, I., Yamashita, M.: Formations for fast locomotion of metamorphic robotic systems. Int. J. Robot. Res. 23(6), 583–593 (2004)

    Article  Google Scholar 

  23. Dumitrescu, A., Suzuki, I., Yamashita, M.: Motion planning for metamorphic systems: feasibility, decidability, and distributed reconfiguration. IEEE Trans. Robot. Autom. 20(3), 409–418 (2004)

    Article  Google Scholar 

  24. Fekete, S., Gmyr, R., Hugo, S., Keldenich, P., Scheffer, C., Schmidt, A.: CADbots: algorithmic aspects of manipulating programmable matter with finite automata. CoRR abs/1810.06360 (2018)

    Google Scholar 

  25. Fekete, S., Richa, A., Römer, K., Scheideler, C.: Algorithmic foundations of programmable matter (Dagstuhl Seminar 16271). In: Dagstuhl Reports, vol. 6 (2016). Also in ACM SIGACT News, 48(2), 87–94 (2017)

    Google Scholar 

  26. Gilpin, K., Knaian, A., Rus, D.: Robot pebbles: one centimeter modules for programmable matter through self-disassembly. In: 2010 IEEE International Conference on Robotics and Automation (ICRA), pp. 2485–2492. IEEE (2010)

    Google Scholar 

  27. Gmyr, R., et al.: Forming tile shapes with simple robots. Nat. Comput. 19(2), 375–390 (2019). https://doi.org/10.1007/s11047-019-09774-2

    Article  MathSciNet  Google Scholar 

  28. Itai, A., Papadimitriou, C., Szwarcfiter, J.: Hamilton paths in grid graphs. SIAM J. Comput. 11(4), 676–686 (1982)

    Article  MathSciNet  Google Scholar 

  29. Knaian, A., Cheung, K., Lobovsky, M., Oines, A., Schmidt-Neilsen, P., Gershenfeld, N.: The Milli-Motein: a self-folding chain of programmable matter with a one centimeter module pitch. In: 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1447–1453. IEEE (2012)

    Google Scholar 

  30. Michail, O., Skretas, G., Spirakis, P.: On the transformation capability of feasible mechanisms for programmable matter. J. Comput. Syst. Sci. 102, 18–39 (2019)

    Article  MathSciNet  Google Scholar 

  31. Michail, O., Spirakis, P.G.: Simple and efficient local codes for distributed stable network construction. Distrib. Comput. 29(3), 207–237 (2015). https://doi.org/10.1007/s00446-015-0257-4

    Article  MathSciNet  MATH  Google Scholar 

  32. Michail, O., Spirakis, P.: Elements of the theory of dynamic networks. Commun. ACM 61(2), 72–81 (2018)

    Article  Google Scholar 

  33. Nguyen, A., Guibas, L., Yim, M.: Controlled module density helps reconfiguration planning. In: Proceedings of 4th International Workshop on Algorithmic Foundations of Robotics, pp. 23–36 (2000)

    Google Scholar 

  34. Rothemund, P.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)

    Article  Google Scholar 

  35. Rothemund, P., Winfree, E.: The program-size complexity of self-assembled squares. In: Proceedings of the 32nd annual ACM symposium on Theory of computing (STOC), pp. 459–468. ACM (2000)

    Google Scholar 

  36. Rubenstein, M., Cornejo, A., Nagpal, R.: Programmable self-assembly in a thousand-robot swarm. Science 345(6198), 795–799 (2014)

    Article  Google Scholar 

  37. Walter, J., Welch, J., Amato, N.: Distributed reconfiguration of metamorphic robot chains. Distrib. Comput. 17(2), 171–189 (2004)

    Article  Google Scholar 

  38. Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998

    Google Scholar 

  39. Woods, D., Chen, H., Goodfriend, S., Dabby, N., Winfree, E., Yin, P.: Active self-assembly of algorithmic shapes and patterns in polylogarithmic time. In: Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pp. 353–354. ACM (2013)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  41. Yamauchi, Y., Uehara, T., Yamashita, M.: Brief announcement: pattern formation problem for synchronous mobile robots in the three dimensional Euclidean space. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, pp. 447–449. ACM (2016)

    Google Scholar 

  42. Yim, M., et al.: Modular self-reconfigurable robot systems [grand challenges of robotics]. IEEE Robot. Autom. Mag. 14(1), 43–52 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Abdullah Almethen, Othon Michail & Igor Potapov

Authors
  1. Abdullah Almethen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Othon Michail
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Igor Potapov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Abdullah Almethen , Othon Michail or Igor Potapov .

Editor information

Editors and Affiliations

  1. Mathematics and Computer Science, University of Perugia, Perugia, Italy

    Cristina M. Pinotti

  2. Mathematics and Computer Science, University of Perugia, Perugia, Italy

    Alfredo Navarra

  3. Computer Science and Engineering, Indian Institute of Technology Delhi, New Delhi, India

    Amitabha Bagchi

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Almethen, A., Michail, O., Potapov, I. (2020). On Efficient Connectivity-Preserving Transformations in a Grid. In: Pinotti, C.M., Navarra, A., Bagchi, A. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2020. Lecture Notes in Computer Science(), vol 12503. Springer, Cham. https://doi.org/10.1007/978-3-030-62401-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-62401-9_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-62400-2

  • Online ISBN: 978-3-030-62401-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation