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A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves

Zachary Abel, Hugo A. Akitaya, Scott Duke Kominers, Matias Korman, Frederick Stock

TL;DR

The first universal reconfiguration algorithm is introduced -- i.e., any $n-module configuration can reconfigure itself into any specified $n-module configuration using just sliding moves, making it asymptotically tight.

Abstract

In the modular robot reconfiguration problem, we are given $n$ cube-shaped modules (or robots) as well as two configurations, i.e., placements of the $n$ modules so that their union is face-connected. The goal is to find a sequence of moves that reconfigures the modules from one configuration to the other using "sliding moves," in which a module slides over the face or edge of a neighboring module, maintaining connectivity of the configuration at all times. For many years it has been known that certain module configurations in this model require at least $Ω(n^2)$ moves to reconfigure between them. In this paper, we introduce the first universal reconfiguration algorithm -- i.e., we show that any $n$-module configuration can reconfigure itself into any specified $n$-module configuration using just sliding moves. Our algorithm achieves reconfiguration in $O(n^2)$ moves, making it asymptotically tight. We also present a variation that reconfigures in-place, it ensures that throughout the reconfiguration process, all modules, except for one, will be contained in the union of the bounding boxes of the start and end configuration.

A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves

TL;DR

The first universal reconfiguration algorithm is introduced -- i.e., any n-module configuration using just sliding moves, making it asymptotically tight.

Abstract

In the modular robot reconfiguration problem, we are given cube-shaped modules (or robots) as well as two configurations, i.e., placements of the modules so that their union is face-connected. The goal is to find a sequence of moves that reconfigures the modules from one configuration to the other using "sliding moves," in which a module slides over the face or edge of a neighboring module, maintaining connectivity of the configuration at all times. For many years it has been known that certain module configurations in this model require at least moves to reconfigure between them. In this paper, we introduce the first universal reconfiguration algorithm -- i.e., we show that any -module configuration can reconfigure itself into any specified -module configuration using just sliding moves. Our algorithm achieves reconfiguration in moves, making it asymptotically tight. We also present a variation that reconfigures in-place, it ensures that throughout the reconfiguration process, all modules, except for one, will be contained in the union of the bounding boxes of the start and end configuration.

Paper Structure

This paper contains 14 sections, 18 theorems, 1 equation, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries: The Sliding Cube Model
  3. Reconfiguration Framework
  4. Structural Properties
  5. Reconfiguring into a Canonical Configuration
  6. LocateAndFree-based Algorithm
  7. Algorithm Analysis
  8. Bounded-Space Algorithm
  9. Algorithm Overview
  10. Scaffolding Phase
  11. Meld Phase
  12. Reconfiguration to Compact form
  13. Removing Assumptions
  14. Conclusions and Future Work

Key Result

Theorem 1

Given any two connected unlabeled configurations $C$ and $C'$ each having $n\ge 2$ modules, there exists a reconfiguration of $C$ into $C'$ using $O(n^2)$ sliding moves.

Figures (11)

  • Figure 1: Two types of allowed moves in the sliding cube model
  • Figure 2: Examples of slide adjacency. In (a) and (b), $f$ and $f'$ are slide-adjacent; (c) shows how two faces $f$ and $f'$ can share an edge but not be slide-adjacent.
  • Figure 3: Lemma \ref{['lem:structure']}: removing $\mathbf x$ (black) disconnects $\mathbf y$ (dark gray) from the boundary $B_\mathrm{out}(C)$. (Two views are presented.)
  • Figure 4: For a face $f$, with four neighbors $f_\uparrow$, $f_\rightarrow$, $f_\downarrow$, and $f_\leftarrow$, there are four cycles---one per vertex $v$---which visit $f$ and its two neighbors that are adjacent to $v$.
  • Figure 5: $O_i$ and $I_i$ are connected by one module; $O_{i+1}$ and $I_{i+1}$ are a subset of $I_{i}$.
  • ...and 6 more figures

Theorems & Definitions (35)

  • Theorem 1
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 25 more