Universal Coating by 3D Hybrid Programmable Matter

TL;DR

This work tackles the 3D coating problem for hybrid programmable matter by modeling a surface as the triangulated graph $G_{\triangle}$ embedded on a 3D object and using rhombic dodecahedron tiles. A finite-state agent with constant memory, carrying at most one tile, coats the surface in $O(n^2)$ steps with a single tile type, by exploring a boundary path with a left-right rule and maintaining connectivity through carefully defined links, segments, and a tunnel structure. To handle broader object classes, the authors construct a virtual graph $G_{\triangle}^*$ and show how the coating algorithm can be emulated using $2^{2\Delta}$ tile types, incurring $O(\Delta^2 n^2)$ time, with $\Delta$ constant in the model; this establishes worst-case optimality while bridging 2D and 3D hybrid models. The findings have implications for scalable, programmable coating at the nanoscale, potentially enabling medical applications where autonomous particles coat complex 3D surfaces, and they open questions about the minimal tile-type requirements and tighter optimality bounds. All mathematical expressions are presented with proper delimiters, such as $O(n^2)$ and $2^{2\Delta}$, to ensure precise communication of complexity and construction details.

Abstract

Motivated by the prospect of nano-robots that assist human physiological functions at the nanoscale, we investigate the coating problem in the three-dimensional model for hybrid programmable matter. In this model, a single agent with strictly limited viewing range and the computational capability of a deterministic finite automaton can act on passive tiles by picking up a tile, moving, and placing it at some spot. The goal of the coating problem is to fill each node of some surface graph of size $n$ with a tile. We first solve the problem on a restricted class of graphs with a single tile type, and then use constantly many tile types to encode this graph in certain surface graphs capturing the surface of 3D objects. Our algorithm requires $\mathcal{O}(n^2)$ steps, which is worst-case optimal compared to an agent with global knowledge and no memory restrictions.

Figures (8)

• Figure 1: Tiled nodes of the underlying graph $G$ and their incident edges to empty nodes.
• Figure 2: A passive tile (rhombic dodecahedron) and the twelve compass directions.
• Figure 3: (a) Example of a triangulation $G_{\!\triangle}{}$. (b) The simple path $\tau$ that starts at the material depot (hexagon) along the boundary of tiled nodes (opaque surface area). Links are depicted as circles. The blue area ($i$) is the range explored by the agent. In the yellow area ($ii$), $\tau$ is 'exposed' to nodes that are not tiled and not links. The agent explores the blue area to find an overlap with the yellow area, in which case a link can be tiled while preserving connectivity of $G_{\!\triangle}(\mathcal{E})$.
• Figure 4: Tiled nodes are depicted as hexagons, empty nodes as disks, links are depicted in red (node $\sigma_i$ in (b), $q$ in (c) and the uppermost central node in (c)). (a) Example of the updates made to the agent's anchor $a(p)$ while following the LHR and RHR. (b) $\textsc{seg}(\sigma_i), \textsc{tail}(\sigma_i)$ and $\textsc{head}(\sigma_i)$ for some node $\sigma_i \in \sigma$ in an example configuration. (c) Example of the $i$-range $R_i(p,q)$ for $i = 2$ (orange, opaque) together with $N_2(p)$ w.r.t. $G_{\!\triangle}(\mathcal{E}{} \setminus \{q\})$ (green, transparent).
• Figure 5: Examples in which the next tile is placed at some link (red disk) in phase Coat: $\sigma_{i-1}$ is tiled in (a); $\sigma_{i-2}$ is tiled in (b) and (c). Only in (c), ${\textsc{skip}}$ is set to $true$ since $\sigma_{i-3} \notin \textsc{li} \cup \{s^0\}$. As a result, $\sigma_{i-3}$ is tiled on the next visit.

Theorems & Definitions (15)

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