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

Matthias Artmann, Andreas Padalkin, Christian Scheideler

TL;DR

This paper considers 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.

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 $Ω(\sqrt{n})$ 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.

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

TL;DR

This paper considers 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.

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 and have to find maximally scaled representations of 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 synchronous rounds for the general problem, where 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 be the maximum scale of the considered shape in a given amoebot structure. If the shape is star convex, we solve it within rounds. If it is a snowflake but not star convex, we solve it within rounds.

Paper Structure

This paper contains 23 sections, 28 theorems, 4 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometric Amoebot Model
  3. Reconfigurable Circuit Extension
  4. Problem Statement
  5. Related Work
  6. Preliminaries
  7. Coordination and Synchronization
  8. Chains and Chain Primitives
  9. Binary Operations
  10. The PASC Algorithm
  11. A Simple Lower Bound
  12. Helper Procedures
  13. Scale Factor Search
  14. Primitive Shapes
  15. Stretched Shapes
  16. ...and 8 more sections

Key Result

Lemma 1

Let $C = (p_0, \ldots, p_{m-1})$ be an amoebot chain such that each amoebot $p_i$ stores two bits $a_i$ and $b_i$ of the integers $a$ and $b$, where $a = \sum\limits_{i = 0}^{m-1} a_i 2^i$ and $b = \sum\limits_{i = 0}^{m-1} b_i 2^i$. Within $\mathcal{O}\,( 1 )$ rounds, the amoebots on $C$ can compar

Figures (5)

  • Figure 1: Amoebot structure.
  • Figure 2: Reconfigurable circuit extension.
  • Figure 3: Grid axes and cardinal directions.
  • Figure 5: Examples of equivalent and scaled shapes. Each shape is identified by the grid nodes, edges and faces it contains. The origin of each shape is highlighted in white (we place the shapes at different locations for convenience). $S_1$ and $S_2$ are equivalent and each contains one face and one hole. The shapes $S_3$, $S_4$ and $S_5$ illustrate the scaling operation.
  • Figure 8: Examples of snowflakes, star convex shapes and other, non-snowflake shapes. Green nodes indicate star convex shape centers, blue nodes indicate possible snowflake origins and chosen origins are highlighted with a white center. All center nodes are also snowflake origins. Shape (a) is convex and shape (b) demonstrates that not all snowflake origins must be center nodes. Shape (c) is a union of lines, (d) is the Minkowski sum of (c) and $\mathrm{L}(\textsc{E}, 1)$, (e) is the union of $\mathrm{L}(\textsc{E}, 2)$ and $\mathrm{(d)} + 2 \cdot \bm{u}_{ \textsc{E} }$, and shape (f) consists of six rotated copies of (e). (g) is the example shape for the lower bound from Section \ref{['sec:lower_bound']}.

Theorems & Definitions (38)

  • Lemma 1
  • Lemma 2
  • Lemma 3: feldmann2022coordinatingpadalkin2022structural
  • Lemma 4
  • Theorem 5
  • Lemma 6
  • Definition 7
  • Lemma 8
  • Lemma 9
  • Definition 10
  • ...and 28 more