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



Connor, Matthew, Michail, Othon ORCID: 0000-0002-6234-3960 and Potapov, Igor
(2021) Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach. Algorithms for Sensor Systems.

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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.

Item Type: Article
Additional Information: 22 pages, 10 figures
Uncontrolled Keywords: cs.DC, cs.DC
Divisions: Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science
Depositing User: Symplectic Admin
Date Deposited: 02 Sep 2021 08:15
Last Modified: 22 Jul 2022 16:10
DOI: 10.1007/978-3-030-89240-1_4
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3135611