Fast Reconfiguration for Programmable Matter

TL;DR

The paper tackles global shape reconfiguration for programmable matter in the amoebot model, where particles have extremely limited memory and communication. It introduces a non-canonical reconfiguration approach based on feather trees and efficient shortest-path trees to route supply from the initial-to-target symmetric difference through a core, enabling parallel shortest-path movements. The authors prove safety and liveness, establish a worst-case linear time bound of $O(n)$ rounds under a sequential scheduler, and show an $O(n)$-expected (and $O(n\log n)$ w.h.p.) bound under asynchronous scheduling with randomness. The framework naturally accommodates practical advantages when the symmetric difference is small and provides a robust method for shape repair, including extensions to holes and non-simply connected cores, with a coarse-grid crossing scheme that supports scalable parallelism.

Abstract

The concept of programmable matter envisions a very large number of tiny and simple robot particles forming a smart material. Even though the particles are restricted to local communication, local movement, and simple computation, their actions can nevertheless result in the global change of the material's physical properties and geometry. A fundamental algorithmic task for programmable matter is to achieve global shape reconfiguration by specifying local behavior of the particles. In this paper we describe a new approach for shape reconfiguration in the \emph{amoebot} model. The amoebot model is a distributed model which significantly restricts memory, computing, and communication capacity of the individual particles. Thus the challenge lies in coordinating the actions of particles to produce the desired behavior of the global system. Our reconfiguration algorithm is the first algorithm that does not use a canonical intermediate configuration when transforming between arbitrary shapes. We introduce new geometric primitives for amoebots and show how to reconfigure particle systems, using these primitives, in a linear number of activation rounds in the worst case. In practice, our method exploits the geometry of the symmetric difference between input and output shape: it minimizes unnecessary disassembly and reassembly of the particle system when the symmetric difference between the initial and the target shapes is small. Furthermore, our reconfiguration algorithm moves the particles over as many parallel shortest paths as the problem instance allows.

Figures (16)

• Figure 1: Top: particles with ports labeled, in contracted and expanded state. Bottom: handover operation between two particles.
• Figure 2: The particles form the initial shape $I$. The target shape $T$ is shaded in gray. Supply particles are blue, supply roots dark blue. Demand roots (red) store spanning trees of their demand components. The graph $G_L$ on the coarse grid is shown in green and leader $\ell$ is marked. Particles on $G_L$ that are green are grid nodes, other particles on $G_L$ are edge nodes.
• Figure 3: Shortest path map of node $r$. Any shortest path between $r$ and $p$ must pass through the roots of the respective SPM regions ($u$, $w$, and $r_i$). The region $R_0$ (in purple) consists of the particles $\mathcal{P}\xspace$-visible to $r$. The red and the blue particles are the roots of the corresponding SPM regions. Right: any path not going through the root of a visibility region can be shortened.
• Figure 4: Left: An angle monotone path from $r$ to $p$. For every particle $q$, the remainder of the path lies in a $60^\circ$-cone. Middle: Growing an SP-tree using cones of directions. The particle on the left just extended its cone to $180^\circ$. Right: A couple activations later.
• Figure 5: Two feather trees growing from the dark blue root. Shafts are red and branches are blue. Left: every particle is reachable by the initial feathers; Right: additional feathers are necessary.

Theorems & Definitions (58)

• Lemma 1: boulinier2008space
• Lemma 2
• proof
• Corollary 3
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• Lemma 7