Document Open Access Logo

Sliding Squares in Parallel

Authors Hugo A. Akitaya , Sándor P. Fekete , Peter Kramer , Saba Molaei , Christian Rieck , Frederick Stock , Tobias Wallner

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.28.pdf
  • Filesize: 1.14 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Computing methodologies → Motion path planning
Keywords
  • Sliding squares
  • parallel motion
  • reconfigurability
  • motion planning
  • multi-agent path finding
  • makespan
  • swarm robotics
  • computational geometry

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We consider algorithmic problems motivated by modular robotic reconfiguration in the sliding square model, in which we are given n square-shaped modules in a (labeled or unlabeled) start configuration and need to find a schedule of sliding moves to transform it into a desired goal configuration, maintaining connectivity of the configuration at all times. Recent work has aimed at minimizing the total number of moves, resulting in fully sequential schedules that can perform reconfiguration in 𝒪(n²) moves, or 𝒪(nP) for arrangements of bounding box perimeter size P.
We provide first results in the sliding square model that exploit parallel motion, performing reconfiguration in worst-case optimal makespan of 𝒪(P). We also provide tight bounds on the complexity of the problem by showing that even deciding the possibility of reconfiguration within makespan 1 is NP-complete in the unlabeled case. In the labeled variant, we note that deciding the same for makespan 2 is NP-complete, while makespan 1 is straightforward.

Cite As Get BibTex

Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner. Sliding Squares in Parallel. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.28

Author Details

Hugo A. Akitaya
  • Miner School of Computer and Information Sciences, University of Massachusetts Lowell, MA, USA
Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Germany
Peter Kramer
  • Department of Computer Science, TU Braunschweig, Germany
Saba Molaei
  • Sharif University of Technology, Teheran, Iran
Christian Rieck
  • Department of Discrete Mathematics, University of Kassel, Germany
Frederick Stock
  • Miner School of Computer and Information Sciences, University of Massachusetts Lowell, MA, USA
Tobias Wallner
  • Department of Computer Science, TU Braunschweig, Germany

Funding

This work was partially supported by the German Research Foundation (DFG), project "Space Ants" (FE 407/22-1) and grant 522790373, by the National Science Foundation (NSF), grant CCF-2348067, and by a fellowship of the German Academic Exchange Service (DAAD).

References

  1. Zachary Abel, Hugo A. Akitaya, Scott Duke Kominers, Matias Korman, and Frederick Stock. A universal in-place reconfiguration algorithm for sliding cube-shaped robots in a quadratic number of moves. In Symposium on Computational Geometry (SoCG), pages 1:1-1:14, 2024. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.1.
  2. Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmovic, Robin Y. Flatland, Matias Korman, Belén 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, 2021. URL: https://doi.org/10.1007/s00453-020-00784-6.
  3. Hugo A. Akitaya, Erik D. Demaine, Andrei Gonczi, Dylan H. Hendrickson, Adam Hesterberg, Matias Korman, Oliver Korten, Jayson Lynch, Irene Parada, and Vera Sacristán. Characterizing universal reconfigurability of modular pivoting robots. In Symposium on Computational Geometry (SoCG), pages 10:1-10:20, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.10.
  4. Hugo A. Akitaya, Erik D. Demaine, Matias Korman, Irina Kostitsyna, Irene Parada, Willem Sonke, Bettina Speckmann, Ryuhei Uehara, and Jules Wulms. Compacting squares: Input-sensitive in-place reconfiguration of sliding squares. In Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 4:1-4:19, 2022. URL: https://doi.org/10.4230/LIPIcs.SWAT.2022.4.
  5. Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner. Sliding squares in parallel, 2025. URL: https://doi.org/10.48550/arXiv.2412.05523.
  6. Greg Aloupis, Sébastien Collette, Mirela Damian, Erik D. Demaine, Robin Flatland, Stefan Langerman, Joseph O'Rourke, Suneeta Ramaswami, Vera Sacristán, and Stefanie Wuhrer. Linear reconfiguration of cube-style modular robots. Computational Geometry, 42(6-7):652-663, 2009. URL: https://doi.org/10.1016/J.COMGEO.2008.11.003.
  7. Aaron T. Becker, Sándor P. Fekete, Phillip Keldenich, Matthias Konitzny, Lillian Lin, and Christian Scheffer. Coordinated motion planning: The video. In Symposium on Computational Geometry (SoCG), pages 74:1-74:6, 2018. URL: https://doi.org/10.4230/LIPICS.SOCG.2018.74.
  8. Julien Bourgeois, Sándor P. Fekete, Ramin Kosfeld, Peter Kramer, Benoît Piranda, Christian Rieck, and Christian Scheffer. Space ants: Episode II - Coordinating connected catoms. In Symposium on Computational Geometry (SoCG), pages 65:1-65:6, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.65.
  9. Zack Butler and Daniela Rus. Distributed planning and control for modular robots with unit-compressible modules. The International Journal of Robotics Research, 22(9):699-715, 2003. URL: https://doi.org/10.1177/02783649030229002.
  10. Matthew Connor, Othon Michail, and George Skretas. All for one and one for all: An 𝒪(1)-musketeers universal transformation for rotating robots. In Symposium on Algorithmic Foundations of Dynamic Networks (SAND), pages 9:1-9:20, 2024. URL: https://doi.org/10.4230/LIPIcs.SAND.2024.9.
  11. Mark de Berg and Amirali Khosravi. Optimal binary space partitions for segments in the plane. International Journal on Computational Geometry and Applications, 22(3):187-206, 2012. URL: https://doi.org/10.1142/S0218195912500045.
  12. Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich, Henk Meijer, and Christian Scheffer. Coordinated motion planning: Reconfiguring a swarm of labeled robots with bounded stretch. SIAM Journal on Computing, 48(6):1727-1762, 2019. URL: https://doi.org/10.1137/18M1194341.
  13. Adrian Dumitrescu and János Pach. Pushing squares around. Graphs and Combinatorics, 22(1):37-50, 2006. URL: https://doi.org/10.1007/s00373-005-0640-1.
  14. Adrian Dumitrescu, Ichiro Suzuki, and Masafumi Yamashita. Formations for fast locomotion of metamorphic robotic systems. International Journal of Robotics Research, 23(6):583-593, 2004. URL: https://doi.org/10.1177/0278364904039652.
  15. Adrian Dumitrescu, Ichiro Suzuki, and Masafumi Yamashita. Motion planning for metamorphic systems: feasibility, decidability, and distributed reconfiguration. Transactions on Robotics, 20(3):409-418, 2004. URL: https://doi.org/10.1109/TRA.2004.824936.
  16. 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, 2023. URL: https://doi.org/10.1007/S10458-023-09626-5.
  17. Sándor P. Fekete, Ramin Kosfeld, Peter Kramer, Jonas Neutzner, Christian Rieck, and Christian Scheffer. Coordinated motion planning: Multi-agent path finding in a densely packed, bounded domain. In International Symposium on Algorithms and Computation (ISAAC), pages 29:1-29:15, 2024. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2024.29.
  18. Sándor P. Fekete, Peter Kramer, Christian Rieck, Christian Scheffer, and Arne Schmidt. Efficiently reconfiguring a connected swarm of labeled robots. Autonomous Agents and Multi-Agent Systems, 38(2):39, 2024. URL: https://doi.org/10.1007/S10458-024-09668-3.
  19. Robert Fitch, Zack J. Butler, and Daniela Rus. Reconfiguration planning for heterogeneous self-reconfiguring robots. In International Conference on Intelligent Robots and Systems (IROS), pages 2460-2467, 2003. URL: https://doi.org/10.1109/IROS.2003.1249239.
  20. Toshio Fukuda, Seiya Nakagawa, Yoshio Kawauchi, and Martin Buss. Structure decision method for self organising robots based on cell structures-cebot. In International Conference on Robotics and Automation (ICRA), pages 695-696, 1989. URL: https://doi.org/10.1109/ROBOT.1989.100066.
  21. Ferran Hurtado, Enrique Molina, Suneeta Ramaswami, and Vera Sacristán Adinolfi. Distributed reconfiguration of 2d lattice-based modular robotic systems. Autonomous Robots, 38(4):383-413, 2015. URL: https://doi.org/10.1007/S10514-015-9421-8.
  22. Irina Kostitsyna, Tim Ophelders, Irene Parada, Tom Peters, Willem Sonke, and Bettina Speckmann. Optimal in-place compaction of sliding cubes. In Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 31:1-31:14, 2024. URL: https://doi.org/10.4230/LIPICS.SWAT.2024.31.
  23. Keith D. Kotay and Daniela L. Rus. Algorithms for self-reconfiguring molecule motion planning. In International Conference on Intelligent Robots and Systems (IROS), pages 2184-2193, 2000. URL: https://doi.org/10.1109/IROS.2000.895294.
  24. 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, 2019. URL: https://doi.org/10.1016/j.jcss.2018.12.001.
  25. Joel Moreno and Vera Sacristán. Reconfiguring sliding squares in-place by flooding. In European Workshop on Computational Geometry (EuroCG), pages 32:1-32:7, 2020. URL: https://www1.pub.informatik.uni-wuerzburg.de/eurocg2020/data/uploads/papers/eurocg20_paper_32.pdf.
  26. An Nguyen, Leonidas J. Guibas, and Mark Yim. Controlled module density helps reconfiguration planning. New Directions in Algorithmic and Computational Robotics, pages 23-36, 2001. Google Scholar
  27. Irene Parada, Vera Sacristán, and Rodrigo I. Silveira. A new meta-module design for efficient reconfiguration of modular robots. Autonomous Robots, 45(4):457-472, 2021. URL: https://doi.org/10.1007/S10514-021-09977-6.
  28. Daniela Rus and Marsette Vona. Crystalline robots: Self-reconfiguration with compressible unit modules. Autonomous Robots, 10:107-124, 2001. URL: https://doi.org/10.1023/A:1026504804984.
  29. Jacob T. Schwartz and Micha Sharir. On the piano movers' problem: III. Coordinating the motion of several independent bodies: the special case of circular bodies moving amidst polygonal barriers. International Journal of Robotics Research, 2(3):46-75, 1983. URL: https://doi.org/10.1177/027836498300200304.
  30. Serguei Vassilvitskii, Mark Yim, and John Suh. A complete, local and parallel reconfiguration algorithm for cube style modular robots. In International Conference on Robotics and Automation (ICRA), pages 117-122, 2002. URL: https://doi.org/10.1109/ROBOT.2002.1013348.
  31. Matthijs Wolters. Parallel algorithms for sliding squares. Master’s thesis, Utrecht University, 2024. Google Scholar
  32. Mark Yim, Wei-Min Shen, Behnam Salemi, Daniela Rus, Mark Moll, Hod Lipson, Eric Klavins, and Gregory S. Chirikjian. Modular self-reconfigurable robot systems [grand challenges of robotics]. Robotics and Automation Magazine, 14(1):43-52, 2007. URL: https://doi.org/10.1109/MRA.2007.339623.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact