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Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach

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Algorithms for Sensor Systems (ALGOSENSORS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 12961))

Abstract

We study a model of programmable matter systems consisting of n devices lying on a 2-dimensional square grid which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We investigate the class of shapes which can be constructed in such a scenario under the additional constraint of maintaining global connectivity at all times. We focus on the scenario of transforming nice shapes, a class of shapes consisting of a central line L where for all nodes u in S either \(u \in L\) or u is connected to L by a line of nodes perpendicular to L. We prove that by introducing a minimal 3-node seed it is possible for the canonical shape of a line of n nodes to be transformed into a nice shape of \(n-1\) nodes. We use this to show that a 4-node seed enables the transformation of nice shapes of size n into any other nice shape of size n in \(O(n^2)\) time. We leave as an open problem the expansion of the class of shapes which can be constructed using such a seed to include those derived from nice shapes.

The full version of the paper with all omitted details is available on arXiv at: https://arxiv.org/abs/2108.09250 .

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Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Matthew Connor, Othon Michail & Igor Potapov

Authors
  1. Matthew Connor
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  2. Othon Michail
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  3. Igor Potapov
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Correspondence to Matthew Connor .

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Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Leszek Gąsieniec

  2. CNRS and University of Bordeaux, Talence, France

    Ralf Klasing

  3. King's College London, London, UK

    Tomasz Radzik

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Connor, M., Michail, O., Potapov, I. (2021). Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach. In: Gąsieniec, L., Klasing, R., Radzik, T. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2021. Lecture Notes in Computer Science(), vol 12961. Springer, Cham. https://doi.org/10.1007/978-3-030-89240-1_4

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  • DOI: https://doi.org/10.1007/978-3-030-89240-1_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-89239-5

  • Online ISBN: 978-3-030-89240-1

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