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Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers

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Abstract

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses \(O(n^2)\) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.

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Notes

  1. The Three Musketeers is a story about four musketeers. This paper is a story about five musketeers.

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Acknowledgements

This research started at the 32nd Bellairs Winter Workshop on Computational Geometry in 2017. We want to thank all participants for the fruitful discussions on the topic. We would also like to thank the reviewers for their insightful and valuable comments.

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Authors and Affiliations

  1. Carleton University, Ottawa, Canada

    Hugo A. Akitaya

  2. State University of New York at Stony Brook, Stony Brook, USA

    Esther M. Arkin

  3. Villanova University, Villanova, USA

    Mirela Damian

  4. Massachusetts Institute of Technology, Cambridge, USA

    Erik D. Demaine

  5. University of Ottawa, Ottawa, Canada

    Vida Dujmović

  6. Siena College, Loudonville, USA

    Robin Flatland

  7. Tufts University, Medford, USA

    Matias Korman

  8. Universidad de Valladolid, Valladolid, Spain

    Belen Palop

  9. TU Eindhoven, Eindhoven, The Netherlands

    Irene Parada

  10. The University of Sydney, Sydney, Australia

    André van Renssen

  11. Universitat Politècnica de Catalunya, Barcelona, Spain

    Vera Sacristán

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  1. Hugo A. Akitaya
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  8. Belen Palop
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Correspondence to Irene Parada.

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An extended abstract of this paper has been presented at the 27th Annual European Symposium on Algorithms (ESA 2019).

H. A. A. was supported by NSF CCF-1422311 and CCF-1423615. E. A. was partially funded by NSF (CCF-1526406). E. D. D. was supported in part by NSF ODISSEI Grant EFRI-1240383 and NSF Expedition Grant CCF-1138967. B. P. and V. S. were partially supported by MTM2015-63791-R (MINECO/FEDER). I. P. was supported by the Austrian Science Fund (FWF): W1230. A. R. was supported by JST ERATO Grant Number JPMJER1201, Japan. V.S. was also partially supported by Gen. Cat. DGR 2017SGR1640.

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Akitaya, H.A., Arkin, E.M., Damian, M. et al. Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers. Algorithmica 83, 1316–1351 (2021). https://doi.org/10.1007/s00453-020-00784-6

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