Efficient Shape Formation by 3D Hybrid Programmable Matter: An Algorithm for Low Diameter Intermediate Structures

TL;DR

The paper tackles efficient shape formation in 3D hybrid programmable matter by introducing the icicle as a low-diameter intermediate shape and designing a single-agent algorithm to transform any connected tile configuration into an icicle in $O(n^3)$ steps. The method combines a 2D parallelogram formation phase with a 3D icicle formation phase, using fragmentation concepts and projections to preserve connectivity and guarantee convergence. Theoretical analysis via potential functions and extensive simulations demonstrate diameter reduction on average and competitive runtime, with a best-case icicle diameter of $O(n^{1/3})$. This work hence provides a principled intermediate-structure approach to complex 3D self-reconfiguration that can enable more efficient subsequent tasks in programmable matter systems.

Abstract

This paper considers the shape formation problem within the 3D hybrid model, where a single agent with a strictly limited viewing range and the computational capacity of a deterministic finite automaton manipulates passive tiles through pick-up, movement, and placement actions. The goal is to reconfigure a set of tiles into a specific shape termed an icicle. The icicle, identified as a dense, hole-free structure, is strategically chosen to function as an intermediate shape for more intricate shape formation tasks. It is designed for easy exploration by a finite state agent, enabling the identification of tiles that can be lifted without breaking connectivity. Compared to the line shape, the icicle presents distinct advantages, including a reduced diameter and the presence of multiple removable tiles. We propose an algorithm that transforms an arbitrary initially connected tile structure into an icicle in $\mathcal{O}(n^3)$ steps, matching the runtime of the line formation algorithm from prior work. Our theoretical contribution is accompanied by an extensive experimental analysis, indicating that our algorithm decreases the diameter of tile structures on average.

Figures (8)

• Figure 1: (a) An example configuration that has the shape of an icicle; the agent (depicted as a sphere) is positioned at a tiled node within the platform representing a parallelogram. (b--d) The twelve compass directions divided into upwards (b), plane (c) and downwards directions (d).
• Figure 2: Illustrating the $x$-, $y$-, and $z$-coordinate axes (a), the bounding cylinder (b), which infinitely extends in directions ${\textsc{une}}$ and ${\textsc{dsw}}$ as indicated by the arrows, and the bounding box (c) of an example configuration. Tiles are shaded according to their $z$-coordinate, with brighter shades representing lower $z$-coordinates. In the example, there is one layer that contains two fragments (darkest shade of gray), and four layers that each contain a single fragment.
• Figure 3: The parallelogram formation algorithm on a 2D configuration. The agent performs multiple steps between each depicted configuration. In (a) and (b) the agent finds a westernmost column, and in (j) the agent terminates. In all other cases, a tile is shifted from the cross to the circle, where the dashed lines indicate the path traversed before placing the tile. The path back to where the tile is picked up as well as the movement to the next column (e.g., (e)--(f)) is not shown.
• Figure 4: During a projection, the agent (black disk) shifts each tile of a fragment in direction ${\textsc{dsw}}$. Detailed in (a-d) is the projection of a single column; (e) is a snapshot of the configuration after the projection. The special case of a parallelogram with a height of one is shown in (f). To maintain connectivity in that case, the agent moves ${\textsc{sw}} + {\textsc{dnw}}$ to transition below the next column.
• Figure 5: Illustrating all scenarios in which the northernmost node of a locally westernmost column is not removable. For brevity, (a–-r) only illustrate the agent's movement (indicated by arrows) to the empty node (with a dashed outline) that is tiled next; (s) and (t) also portray the subsequent tile shifts. In (r), the agent may alternatively enter BuildPar if the outlined node were tiled. Note that (s) and (t) only show instances where a tile at ${\textsc{dsw}}$ (s) and ${\textsc{de}}$ (t) is encountered.

Theorems & Definitions (14)

• Definition 1
• Lemma 2
• Lemma 3
• Lemma 4
• Definition 5
• Definition 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10