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

Matthew Connor; Othon Michail; Igor Potapov

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arXiv · 2108.09250

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

Matthew Connor·Othon Michail·Igor Potapov

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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.

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BibTeX RIS

Matthew Connor, Othon Michail, Igor Potapov. 2021-08-20. Centralised Connectivity-Preserving Transformations for Programmable Matter: A Minimal Seed Approach. https://arxiv.org/abs/2108.09250

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