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

TL;DR

This paper explores how to transform shapes in programmable matter systems while maintaining connectivity, demonstrating minimal seed configurations that enable shape transformations efficiently.

Contribution

It introduces a minimal seed approach for transforming and constructing nice shapes in programmable matter with connectivity constraints.

Findings

A 3-node seed allows transforming a line into a nice shape of size n-1.

A 4-node seed enables transforming any nice shape into another of the same size in O(n^2) time.

Open problem: extending the class of constructible shapes beyond nice shapes.

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.