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DOI: 10.4230/LIPIcs.SAND.2024.9

All for One and One for All: An O(1)-Musketeers Universal Transformation for Rotating Robots

Authors: Matthew Connor, Othon Michail, and George Skretas

Published in: LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Abstract

In this paper, we settle the main open question of [Michail, Skretas, Spirakis, ICALP'17], asking what is the family of two-dimensional geometric shapes that can be transformed into each other by a sequence of rotation operations, none of which disconnects the shape. The model represents programmable matter systems consisting of interconnected modules that perform the minimal mechanical operation of 90° rotations around each other. The goal is to transform an initial shape of modules A into a target shape B. Under the necessary assumptions that the given shapes are connected and have identical colourings on a checkered colouring of the grid, and using a seed of only constant size, we prove that any pair of such shapes can be transformed into each other within an optimal O(n²) rotation operations none of which disconnects the shape.

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Matthew Connor, Othon Michail, and George Skretas. All for One and One for All: An O(1)-Musketeers Universal Transformation for Rotating Robots. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{connor_et_al:LIPIcs.SAND.2024.9,
  author =	{Connor, Matthew and Michail, Othon and Skretas, George},
  title =	{{All for One and One for All: An O(1)-Musketeers Universal Transformation for Rotating Robots}},
  booktitle =	{3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)},
  pages =	{9:1--9:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-315-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{292},
  editor =	{Casteigts, Arnaud and Kuhn, Fabian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.9},
  URN =		{urn:nbn:de:0030-drops-198874},
  doi =		{10.4230/LIPIcs.SAND.2024.9},
  annote =	{Keywords: programmable matter, universal transformation, reconfigurable robotics, shape formation, centralised algorithms}
}

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All for One and One for All: An O(1)-Musketeers Universal Transformation for Rotating Robots

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