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All for One and One for All: An O(1)-Musketeers Universal Transformation for Rotating Robots

Authors Matthew Connor, Othon Michail , George Skretas

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Author Details

Matthew Connor
  • Department of Computer Science, University of Liverpool, UK
Othon Michail
  • Department of Computer Science, University of Liverpool, UK
George Skretas
  • Hasso Plattner Institute, University of Potsdam, Germany

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Matthew Connor, Othon Michail, and George Skretas. All for One and One for All: An O(1)-Musketeers Universal Transformation for Rotating Robots. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SAND.2024.9

Abstract

In this paper, we settle the main open question of [Michail, Skretas, Spirakis, ICALP'17], asking what is the family of two-dimensional geometric shapes that can be transformed into each other by a sequence of rotation operations, none of which disconnects the shape. The model represents programmable matter systems consisting of interconnected modules that perform the minimal mechanical operation of 90° rotations around each other. The goal is to transform an initial shape of modules A into a target shape B. Under the necessary assumptions that the given shapes are connected and have identical colourings on a checkered colouring of the grid, and using a seed of only constant size, we prove that any pair of such shapes can be transformed into each other within an optimal O(n²) rotation operations none of which disconnects the shape.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • programmable matter
  • universal transformation
  • reconfigurable robotics
  • shape formation
  • centralised algorithms

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References

  1. Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada, André van Renssen, and Vera Sacristán. Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers. Algorithmica, 83(5):1316-1351, May 2021. URL: https://doi.org/10.1007/s00453-020-00784-6.
  2. Nada Almalki and Othon Michail. On geometric shape construction via growth operations. Theoretical Computer Science, 984:114324, February 2024. URL: https://doi.org/10.1016/j.tcs.2023.114324.
  3. Abdullah Almethen, Othon Michail, and Igor Potapov. Pushing lines helps: Efficient Universal Centralised Transformations for Programmable Matter. Theoretical Computer Science, 830-831:43-59, August 2020. URL: https://doi.org/10.1016/j.tcs.2020.04.026.
  4. Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. Computation in Networks of Passively Mobile Finite-State Sensors. Distributed Computing, 18(4):235-253, March 2006. URL: https://doi.org/10.1007/s00446-005-0138-3.
  5. Matthew Connor and Othon Michail. Centralised connectivity-preserving transformations by rotation: 3 musketeers for all orthogonal convex shapes. In Proceedings of the 18th International Symposium on Algorithmics of Wireless Networks (ALGOSENSORS), pages 60-76, 2022. Google Scholar
  6. Matthew Connor, Othon Michail, and Igor Potapov. Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach. Theoretical Computer Science, November 2022. URL: https://doi.org/10.1016/j.tcs.2022.09.016.
  7. Joshua J. Daymude, Zahra Derakhshandeh, Robert Gmyr, Alexandra Porter, Andréa W. Richa, Christian Scheideler, and Thim Strothmann. On the Runtime of Universal Coating for Programmable Matter. Natural Computing, 17(1):81-96, March 2018. URL: https://doi.org/10.1007/s11047-017-9658-6.
  8. Zahra Derakhshandeh, Shlomi Dolev, Robert Gmyr, Andréa W. Richa, Christian Scheideler, and Thim Strothmann. Amoebot - a new model for programmable matter. In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 220-222, 2014. URL: https://doi.org/10.1145/2612669.2612712.
  9. Zahra Derakhshandeh, Robert Gmyr, Andrea W. Richa, Christian Scheideler, and Thim Strothmann. Universal Shape Formation for Programmable Matter. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 289-299, 2016. URL: https://doi.org/10.1145/2935764.2935784.
  10. Zahra Derakhshandeh, Robert Gmyr, Andréa W. Richa, Christian Scheideler, and Thim Strothmann. 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), pages 1-2, 2015. URL: https://doi.org/10.1145/2800795.2800829.
  11. David Doty. Theory of Algorithmic Self-Assembly. Communications of the ACM, 55(12):78-88, December 2012. URL: https://doi.org/10.1145/2380656.2380675.
  12. David Doty. Timing in Chemical Reaction Networks. In Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings, pages 772-784. December 2013. URL: https://doi.org/10.1137/1.9781611973402.57.
  13. Adrian Dumitrescu and János Pach. Pushing squares around. In Proceedings of the Twentieth ACM Annual Symposium on Computational Geometry (SCG), pages 116-123, 2004. Google Scholar
  14. Adrian Dumitrescu, Ichiro Suzuki, and Masafumi Yamashita. Motion planning for metamorphic systems: Feasibility, decidability, and distributed reconfiguration. IEEE Transactions on Robotics and Automation, 20(3):409-418, 2004. Google Scholar
  15. Sándor P. Fekete, Phillip Keldenich, Ramin Kosfeld, Christian Rieck, and Christian Scheffer. Connected coordinated motion planning with bounded stretch. Autonomous Agents and Multi-Agent Systems, 37(2):43, October 2023. Google Scholar
  16. Michael Feldmann, Andreas Padalkin, Christian Scheideler, and Shlomi Dolev. Coordinating Amoebots via Reconfigurable Circuits. Journal of Computational Biology: A Journal of Computational Molecular Cell Biology, 29(4):317-343, April 2022. URL: https://doi.org/10.1089/cmb.2021.0363.
  17. Kyle Gilpin, Ara Knaian, and Daniela Rus. Robot Pebbles: One Centimeter Modules for Programmable Matter through Self-Disassembly. In 2010 IEEE International Conference on Robotics and Automation, pages 2485-2492, 2010. URL: https://doi.org/10.1109/ROBOT.2010.5509817.
  18. Camille Jordan. Cours d'analyse de l'École polytechnique, volume 1. Gauthier-Villars et fils, 1893. Google Scholar
  19. George B. Mertzios, Othon Michail, George Skretas, Paul G. Spirakis, and Michail Theofilatos. The complexity of growing a graph. In 18th International Symposium on Algorithmics of Wireless Networks (ALGOSENSORS), pages 123-137, 2022. Google Scholar
  20. Othon Michail, George Skretas, and Paul G. Spirakis. On the transformation capability of feasible mechanisms for programmable matter. In 44th International Colloquium on Automata, Languages and Programming (ICALP), pages 136:1-136:15, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.136.
  21. Othon Michail, George Skretas, and Paul G. Spirakis. On The Transformation Capability of Feasible Mechanisms for Programmable Matter. Journal of Computer and System Sciences, 102:18-39, June 2019. URL: https://doi.org/10.1016/j.jcss.2018.12.001.
  22. Othon Michail and Paul G. Spirakis. Simple and Efficient Local Codes for Distributed Stable Network Construction. Distributed Computing, 29(3):207-237, June 2016. URL: https://doi.org/10.1007/s00446-015-0257-4.
  23. Matthew J. Patitz. An introduction to tile-based self-assembly and a survey of recent results. Natural Computing, 13:195-224, June 2014. Google Scholar
  24. Paul W. K. Rothemund. Folding DNA to Create Nanoscale Shapes and Patterns. Nature, 440(7082):297-302, March 2006. URL: https://doi.org/10.1038/nature04586.
  25. Paul W. K. Rothemund and Erik Winfree. The Program-size Complexity of Self-Assembled Squares (extended abstract). In Proceedings of the thirty-second Annual ACM Symposium on Theory of Computing (STOC), pages 459-468, May 2000. URL: https://doi.org/10.1145/335305.335358.
  26. Michael Rubenstein, Alejandro Cornejo, and Radhika Nagpal. Programmable Self-assembly in a Thousand-Robot Swarm. Science, 345(6198):795-799, August 2014. URL: https://doi.org/10.1126/science.1254295.
  27. David Soloveichik, Matthew Cook, Erik Winfree, and Jehoshua Bruck. Computation with Finite Stochastic Chemical Reaction Networks. Natural Computing, 7(4):615-633, December 2008. URL: https://doi.org/10.1007/s11047-008-9067-y.
  28. Pierre Thalamy, Benoit Piranda, and Julien Bourgeois. Distributed Self-Reconfiguration using a Deterministic Autonomous Scaffolding Structure. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pages 140-148, 2019. Google Scholar
  29. Pierre Thalamy, Benoît Piranda, and Julien Bourgeois. 3D Coating Self-Assembly for Modular Robotic Scaffolds. In 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 11688-11695, 2020. ISSN: 2153-0866. URL: https://doi.org/10.1109/IROS45743.2020.9341324.
  30. Erik Winfree. Algorithmic Self-assembly of DNA. PhD thesis, California Institute of Technology, June 1998. Google Scholar
  31. Damien Woods, Ho-Lin Chen, Scott Goodfriend, Nadine Dabby, Erik Winfree, and Peng Yin. Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science (ITCS), pages 353-354, 2013. URL: https://doi.org/10.1145/2422436.2422476.
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