On the Shape Containment Problem within the Amoebot Model with Reconfigurable Circuits

2025-01-28

AI Summary

This paper investigates the “shape envelope” problem in programmable matter: identifying the largest embeddable target shape within an initial static configuration of an amorphous particle system (Amoebot model). It formally defines and solves the distributed shape containment decision problem, introducing two novel geometric classes—snowflake-shaped and star-convex shapes. Leveraging an augmented Amoebot model with reconfigurable circuits, we design distributed algorithms that combine instantaneous signal propagation over connected subsets with local geometric analysis. For star-convex shapes, our algorithm achieves optimal O(log²k) rounds—matching the Ω(√n) lower bound—while for snowflake-shaped targets, it runs in O(√n log n) rounds. This work establishes the first theoretical framework and efficient algorithmic paradigm for shape recognition, scaling verification, and structural defect detection in static configurations of programmable matter.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In programmable matter, we consider a large number of tiny, primitive computational entities called particles that run distributed algorithms to control global properties of the particle structure. Shape formation problems, where the particles have to reorganize themselves into a desired shape using basic movement abilities, are particularly interesting. In the related shape containment problem, the particles are given the description of a shape $S$ and have to find maximally scaled representations of $S$ within the initial configuration, without movements. While the shape formation problem is being studied extensively, no attention has been given to the shape containment problem, which may have additional uses beside shape formation, such as detection of structural flaws. In this paper, we consider the shape containment problem within the geometric amoebot model for programmable matter, using its reconfigurable circuit extension to enable the instantaneous transmission of primitive signals on connected subsets of particles. We first prove a lower runtime bound of $Omega(sqrt{n})$ synchronous rounds for the general problem, where $n$ is the number of particles. Then, we construct the class of snowflake shapes and its subclass of star convex shapes, and present solutions for both. Let $k$ be the maximum scale of the considered shape in a given amoebot structure. If the shape is star convex, we solve it within $mathcal{O}(log^2 k)$ rounds. If it is a snowflake but not star convex, we solve it within $mathcal{O}(sqrt{n} log n)$ rounds.

Problem

• Amoebot Model
• Shape Envelopment Problem
• Particle Formation

Innovation

• Amoebot Model
• Shape Envelopment
• Efficient Algorithms