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

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.