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

Authors Kristian Hinnenthal , David Liedtke , Christian Scheideler

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Author Details

Kristian Hinnenthal
  • Paderborn University, Germany
David Liedtke
  • Paderborn University, Germany
Christian Scheideler
  • Paderborn University, Germany

Funding

This work was supported by the DFG Project SCHE 1592/10-1.

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Kristian Hinnenthal, David Liedtke, and Christian Scheideler. Efficient Shape Formation by 3D Hybrid Programmable Matter: An Algorithm for Low Diameter Intermediate Structures. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SAND.2024.15

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 𝒪(n³) 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Programmable Matter
  • Shape Formation
  • 3D Model
  • Finite Automaton

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