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Algorithmica

pp 1–23 | Cite as

Tilt Assembly: Algorithms for Micro-factories That Build Objects with Uniform External Forces

  • Aaron T. Becker
  • Sándor P. Fekete
  • Phillip Keldenich
  • Dominik Krupke
  • Christian Rieck
  • Christian Scheffer
  • Arne Schmidt
Article
Part of the following topical collections:
  1. Special Issue: Algorithms and Computation

Abstract

We present algorithmic results for the parallel assembly of many micro-scale objects in two and three dimensions from tiny particles, which has been proposed in the context of programmable matter and self-assembly for building high-yield micro-factories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle. Particles bond when forced together with another appropriate particle. Due to the physical and geometric constraints, not all shapes can be built in this manner; this gives rise to the Tilt Assembly Problem (TAP) of deciding constructibility. For simply-connected polyominoes P in 2D consisting of N unit-squares (“tiles”), we prove that TAP can be decided in \(O(N\log N)\) time. For the optimization variant MaxTAP (in which the objective is to construct a subshape of maximum possible size), we show polyAPX-hardness: unless P = NP, MaxTAP cannot be approximated within a factor of \(\Omega (N^{\frac{1}{3}})\); for tree-shaped structures, we give an \(\Omega (N^{\frac{1}{2}})\)-approximation algorithm. For the efficiency of the assembly process itself, we show that any constructible shape allows pipelined assembly, which produces copies of P in O(1) amortized time, i.e., N copies of P in O(N) time steps. These considerations can be extended to three-dimensional objects: For the class of polycubes P we prove that it is NP-hard to decide whether it is possible to construct a path between two points of P; it is also NP-hard to decide constructibility of a polycube P. Moreover, it is expAPX-hard to maximize a sequentially constructible path from a given start point.

Keywords

Programmable matter Micro-factories Tile assembly Tilt Approximation Hardness 

Notes

References

  1. 1.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 740–748 (2001)Google Scholar
  2. 2.
    Arbuckle, D., Requicha, A.A.: Self-assembly and self-repair of arbitrary shapes by a swarm of reactive robots: algorithms and simulations. Auton. Robots 28(2), 197–211 (2010)CrossRefGoogle Scholar
  3. 3.
    Becker, A., Fekete, S.P., Keldenich, P., Konitzny, M., Lillian, L., Scheffer, C.: Coordinated motion planning: the video. In: Proceeedings of the Symposium on Computational Geometry (SoCG), pp. 74:1–74:5 (2018). Video available at http://www.computational-geometry.org
  4. 4.
    Becker, A.T., Demaine, E.D., Fekete, S.P., Habibi, G., McLurkin, J.: Reconfiguring massive particle swarms with limited, global control. In: Proceedings of the International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS), pp. 51–66 (2013)Google Scholar
  5. 5.
    Becker, A.T., Demaine, E.D., Fekete, S.P., McLurkin, J.: Particle computation: designing worlds to control robot swarms with only global signals. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 6751–6756 (2014)Google Scholar
  6. 6.
    Becker, A.T., Ertel, C., McLurkin, J.: Crowdsourcing swarm manipulation experiments: a massive online user study with large swarms of simple robots. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 2825–2830 (2014)Google Scholar
  7. 7.
    Becker, A.T., Felfoul, O., Dupont, P.E.: Simultaneously powering and controlling many actuators with a clinical MRI scanner. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 2017–2023 (2014)Google Scholar
  8. 8.
    Becker, A.T., Felfoul, O., Dupont, P.E.: Toward tissue penetration by MRI-powered millirobots using a self-assembled Gauss gun. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 1184–1189 (2015)Google Scholar
  9. 9.
    Becker, A.T., Habibi, G., Werfel, J., Rubenstein, M., McLurkin, J.: Massive uniform manipulation: controlling large populations of simple robots with a common input signal. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 520–527 (2013)Google Scholar
  10. 10.
    Berman, P., Schnitger, G.: On the complexity of approximating the independent set problem. Inf. Comput. 96(1), 77–94 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cannon, S., Demaine, E.D., Demaine, M.L., Eisenstat, S., Patitz, M.J., Schweller, R., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors). In: Proceedings of the International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 172–184 (2013)Google Scholar
  12. 12.
    Chalk, C., Martinez, E., Schweller, R., Vega, L., Winslow, A., Wylie, T.: Optimal staged self-assembly of general shapes. In: Proceedings of the European Symposium on Algorithms (ESA), pp. 26:1–26:17 (2016)Google Scholar
  13. 13.
    Chen, H.-L., Doty, D.: Parallelism and time in hierarchical self-assembly. SIAM J. Comput. 46(2), 661–709 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Demaine, E.D., Demaine, M.L., Fekete, S.P., Ishaque, M., Rafalin, E., Schweller, R.T., Souvaine, D.L.: Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues. Nat. Comput. 7(3), 347–370 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Demaine, E.D., Fekete, S.P., Keldenich, P., Meijer, H., Scheffer, C.: Coordinated motion planning: reconfiguring a swarm of labeled robots with bounded stretch. In: Proceeedings of the Symposium on Computational Geometry (SoCG), pp. 29:1–29:15 (2018)Google Scholar
  16. 16.
    Demaine, E.D., Fekete, S.P., Scheffer, C., Schmidt, A.: New geometric algorithms for fully connected staged self-assembly. Theor. Comput. Sci. 671, 4–18 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Derakhshandeh, Z., Gmyr, R., Richa, A.W., Scheideler, C., Strothmann, T.: An algorithmic framework for shape formation problems in self-organizing particle systems. In: Proceedings of the Second Annual International Conference on Nanoscale Computing and Communication (NANOCOM), p. 21 (2015)Google Scholar
  18. 18.
    Derakhshandeh, Z., Gmyr, R., Richa, A.W., Scheideler, C., Strothmann, T.: Universal coating for programmable matter. Theor. Comput. Sci. 671, 56–68 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hoffmann, M.: Motion planning amidst movable square blocks: Push-* is NP-hard. In: Canadian Conference on Computational Geometry, pp. 205–210 (2000)Google Scholar
  20. 20.
    Kim, P.S.S., Becker, A.T., Ou, Y., Julius, A.A., Kim, M.J.: Imparting magnetic dipole heterogeneity to internalized iron oxide nanoparticles for microorganism swarm control. J. Nanopart. Res. 17(3), 1–15 (2015)Google Scholar
  21. 21.
    Kim, P.S.S., Becker, A.T., Ou, Y., Kim, M.J., et al.: Swarm control of cell-based microrobots using a single global magnetic field. In: Proceedings of the International Conference on Ubiquitous Robotics and Ambient Intelligence (URAI), pp. 21–26 (2013)Google Scholar
  22. 22.
    Mahadev, A.V., Krupke, D., Reinhardt, J.-M., Fekete, S.P., Becker, A.T.: Collecting a swarm in a grid environment using shared, global inputs. In: Proceedings of the IEEE International Conference on Automation Science and Engineering (CASE), pp. 1231–1236 (2016)Google Scholar
  23. 23.
    Manzoor, S., Sheckman, S., Lonsford, J., Kim, H., Kim, M.J., Becker, A.T.: Parallel self-assembly of polyominoes under uniform control inputs. IEEE Robot. Autom. Lett. 2(4), 2040–2047 (2017)Google Scholar
  24. 24.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 459–468 (2000)Google Scholar
  25. 25.
    Rubenstein, M., Cabrera, A., Werfel, J., Habibi, G., McLurkin, J., Nagpal, R.: Collective transport of complex objects by simple robots: theory and experiments. In: Proceedings of the International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 47–54 (2013)Google Scholar
  26. 26.
    Shad, H.M., Morris-Wright, R., Demaine, E.D., Fekete, S.P., Becker, A.T.: Particle computation: device fan-out and binary memory. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 5384–5389 (2015)Google Scholar
  27. 27.
    Shahrokhi, S., Becker, A.T.: Stochastic swarm control with global inputs. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 421–427 (2015)Google Scholar
  28. 28.
    Thubagere, A.J., Li, W., Johnson, R.F., Chen, Z., Doroudi, S., Lee, Y.L., Izatt, G., Wittman, S., Srinivas, N., Woods, D., Winfree, E., Qian, L.: A cargo-sorting DNA robot. Science 357(6356), eaan6558 (2017)CrossRefGoogle Scholar
  29. 29.
    Werfel, J., Nagpal, R.: Extended stigmergy in collective construction. IEEE Intell. Syst. 21(2), 20–28 (2006)CrossRefGoogle Scholar
  30. 30.
    Werfel, J., Nagpal, R.: Three-dimensional construction with mobile robots and modular blocks. Int. J. Robot. Res. 27(3–4), 463–479 (2008)CrossRefGoogle Scholar
  31. 31.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of HoustonHoustonUSA
  2. 2.Department of Computer ScienceTU BraunschweigBraunschweigGermany

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