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

Authors Matthew Connor, Othon Michail , George Skretas

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Author Details

Matthew Connor
  • Department of Computer Science, University of Liverpool, UK
Othon Michail
  • Department of Computer Science, University of Liverpool, UK
George Skretas
  • Hasso Plattner Institute, University of Potsdam, Germany

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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)
https://doi.org/10.4230/LIPIcs.SAND.2024.9

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • programmable matter
  • universal transformation
  • reconfigurable robotics
  • shape formation
  • centralised algorithms

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