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

Authors Hugo A. Akitaya , Esther M. Arkin , Mirela Damian, Erik D. Demaine , Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada , André van Renssen , Vera Sacristán

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Author Details

Hugo A. Akitaya
  • Tufts University, Medford, MA, USA
Esther M. Arkin
  • State University of New York at Stony Brook, NY, USA
Mirela Damian
  • Villanova University, PA, USA
Erik D. Demaine
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Vida Dujmović
  • University of Ottawa, Canada
Robin Flatland
  • Siena College, Loudonville, NY, USA
Matias Korman
  • Tufts University, Medford, MA, USA
Belen Palop
  • Universidad de Valladolid, Spain
Irene Parada
  • Graz University of Technology, Austria
André van Renssen
  • The University of Sydney, Australia
Vera Sacristán
  • Universitat Politècnica de Catalunya, Barcelona, Spain


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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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. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Reconfiguration
  • geometric algorithm
  • pivoting squares
  • modular robots


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