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A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves

Authors Zachary Abel, Hugo A. Akitaya , Scott Duke Kominers, Matias Korman, Frederick Stock

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • modular reconfigurable robots
  • sliding cube model
  • reconfiguration

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Abstract

In the modular robot reconfiguration problem, we are given n cube-shaped modules (or robots) as well as two configurations, i.e., placements of the n modules so that their union is face-connected. The goal is to find a sequence of moves that reconfigures the modules from one configuration to the other using "sliding moves," in which a module slides over the face or edge of a neighboring module, maintaining connectivity of the configuration at all times.
For many years it has been known that certain module configurations in this model require at least Ω(n²) moves to reconfigure between them. In this paper, we introduce the first universal reconfiguration algorithm - i.e., we show that any n-module configuration can reconfigure itself into any specified n-module configuration using just sliding moves. Our algorithm achieves reconfiguration in O(n²) moves, making it asymptotically tight. We also present a variation that reconfigures in-place, it ensures that throughout the reconfiguration process, all modules, except for one, will be contained in the union of the bounding boxes of the start and end configuration.

Cite As Get BibTex

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 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.1

Author Details

Zachary Abel
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Hugo A. Akitaya
  • University of Massachusetts Lowell, MA, USA
Scott Duke Kominers
  • Harvard University, Cambridge, MA, USA
  • a16z crypto, New York, NY, USA
Matias Korman
  • Siemens Electronic Design Automation, Wilsonville, OR, USA
Frederick Stock
  • University of Massachusetts Lowell, MA, USA

Funding

  • Kominers, Scott Duke: Part of this work was conducted during the Simons Laufer Mathematical Sciences Institute Fall 2023 program on the Mathematics and Computer Science of Market and Mechanism Design, which was supported by the National Science Foundation under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778.

Acknowledgements

The authors would like to thank Maarten Löffler and for his contributions during early discussions as well as the authors of [Irina Kostitsyna et al., 2024] and the anonymous reviewers for their valuable comments. Finally, we would like to thank Kevin Li and Colton Wolk for implementing preliminary versions of the algorithms proposed in this paper.

References

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