Document Open Access Logo

Sliding Cubes in Parallel

Authors Hugo A. Akitaya , Joseph Dorfer , Peter Kramer , Christian Rieck , Gabriel Shahrouzi , Frederick Stock

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.26.pdf
  • Filesize: 10.4 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Sliding squares
  • parallel motion
  • reconfigurability
  • three dimensions
  • constant makespan
  • log-APX-hardness
  • NP-hardness
  • worst-case optimality

Metrics

Abstract

In the classic sliding cube model for programmable matter in three dimensions, the task is to find a reconfiguration sequence between two connected configurations of n indistinguishable unit cube modules by sliding modules along their neighbors' faces. Depending on the objective, this sequence should minimize either the total energy expended (the number of moves) or the total elapsed time (the makespan). We give a number of results for the three-dimensional setting, including (i) the first algorithm that achieves worst-case optimal makespan under parallel motion in three dimensions, (ii) a proof of log-APX-hardness to decide either the optimal makespan or the optimal number of moves, which is the strongest known inapproximability bound in any related model, and (iii) a proof of NP-hardness to decide the optimal makespan under parallel motion, even if the two configurations differ only by one module and the optimal makespan is at most two. Our results strengthen the inapproximability claim from [Hugo A. Akitaya et al., 2022] and answer a question of [Akitaya et al., 2025] in the negative.

Cite As Get BibTex

Hugo A. Akitaya, Joseph Dorfer, Peter Kramer, Christian Rieck, Gabriel Shahrouzi, and Frederick Stock. Sliding Cubes in Parallel. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.26

Author Details

Hugo A. Akitaya
  • Miner School of Computer and Information Sciences, University of Massachusetts Lowell, MA, USA
Joseph Dorfer
  • Institute of Algorithms and Theory, Graz University of Technology, Austria
Peter Kramer
  • Department of Computer Science, TU Braunschweig, Germany
Christian Rieck
  • Institute of Mathematics, University of Kassel, Germany
Gabriel Shahrouzi
  • Miner School of Computer and Information Sciences, University of Massachusetts Lowell, MA, USA
Frederick Stock
  • Miner School of Computer and Information Sciences, University of Massachusetts Lowell, MA, USA

Funding

Hugo A. Akitaya, Gabriel Shahrouzi, and Frederick Stock: Supported by the National Science Foundation (NSF), grant CCF-2348067.
  • Dorfer, Joseph: Austrian Science Fund (FWF) 10.55776/DOC183.
  • Kramer, Peter: Partially funded by a fellowship of the German Academic Exchange Service (DAAD) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 530918134.
  • Rieck, Christian: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 522790373.

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, 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.
  3. Hugo A. Akitaya, Joseph Dorfer, Peter Kramer, Christian Rieck, Gabriel Shahrouzi, and Frederick Stock. Sliding cubes in parallel, 2026. URL: https://doi.org/10.48550/arXiv.2603.08537.
  4. Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner. Sliding squares in parallel. In European Symposium on Algorithms (ESA), pages 28:1-28:17, 2025. URL: https://doi.org/10.4230/LIPIcs.ESA.2025.28.
  5. Hugo A. Akitaya, Matias Korman, and Frederick Stock. Input-sensitive reconfiguration of sliding cubes. In Canadian Conference on Computational Geometry (CCCG), pages 216-222, 2025. URL: https://cccg.ca/proceedings/2025/CCCG2025-Proceedings.pdf#doc5B4.3.
  6. Gruia Călinescu, Adrian Dumitrescu, and János Pach. Reconfigurations in graphs and grids. SIAM Journal on Discrete Mathematics, 22(1):124-138, 2008. URL: https://doi.org/10.1137/060652063.
  7. 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.
  8. 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.
  9. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Symposium on Theory of Computing (STOC), pages 624-633, 2014. URL: https://doi.org/10.1145/2591796.2591884.
  10. 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.
  11. 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.
  12. Sándor P. Fekete, Phillip Keldenich, Ramin Kosfeld, Christian Rieck, and Christian Scheffer. Connected coordinated motion planning with bounded stretch. In International Symposium on Algorithms and Computation (ISAAC), pages 9:1-9:16, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.9.
  13. 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.
  14. 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.
  15. Robert Fitch, Zack J. Butler, and Daniela L. 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.
  16. Ferran Hurtado, Enrique Molina, Suneeta Ramaswami, and Vera Sacristán. 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.
  17. 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.
  18. 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.
  19. 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.
  20. 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://computational-geometry.org/abstracts/eurocg/2020.pdf.
  21. 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
  22. Daniel Ratner and Manfred Warmuth. The (n²-1)-puzzle and related relocation problems. Journal of Symbolic Computation, 10(2):111-137, 1990. URL: https://doi.org/10.1016/S0747-7171(08)80001-6.
  23. Matthijs Wolters. Parallel algorithms for sliding squares. Master’s thesis, Utrecht University, 2024. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact