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

TL;DR

This paper presents the first universal in-place reconfiguration algorithm for sliding cube-shaped modular robots, achieving optimal quadratic move complexity and maintaining connectivity throughout the process.

Contribution

It introduces a universal reconfiguration algorithm for any initial and target configurations of cube-shaped modules, with an optimal move complexity of O(n^2).

Findings

Reconfiguration can be achieved in quadratic moves, matching the lower bound.

The algorithm works universally for any configurations of modules.

A variation ensures modules stay within bounding boxes during reconfiguration.

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 $\Omega(n^2)$ 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.