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DOI: 10.4230/LIPIcs.ESA.2019.3
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, and Vera Sacristán

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

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.
Cite as

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)

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@InProceedings{akitaya_et_al:LIPIcs.ESA.2019.3,
  author =	{Akitaya, Hugo A. and Arkin, Esther M. and Damian, Mirela and Demaine, Erik D. and Dujmovi\'{c}, Vida and Flatland, Robin and Korman, Matias and Palop, Belen and Parada, Irene and van Renssen, Andr\'{e} and Sacrist\'{a}n, Vera},
  title =	{{Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.3},
  URN =		{urn:nbn:de:0030-drops-111247},
  doi =		{10.4230/LIPIcs.ESA.2019.3},
  annote =	{Keywords: Reconfiguration, geometric algorithm, pivoting squares, modular robots}
}
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