Input-Sensitive Reconfiguration of Sliding Cubes

TL;DR

This paper presents a new reconfiguration algorithm for modular robots modeled as sliding cubes, achieving input-sensitive move bounds and constant amortized computation per move, improving efficiency over previous methods.

Contribution

The paper introduces a novel reconfiguration algorithm that combines universal and input-sensitive bounds with constant amortized move computation.

Findings

Reconfigures any two shapes in $O(n^2)$ moves.

Reduces move count to the sum of input configuration coordinates.

Achieves $O(1)$ amortized computation per move.

Abstract

A configuration of $n$ unit-cube-shaped \textit{modules} (or \textit{robots}) is a lattice-aligned placement of the $n$ modules so that their union is face-connected. The reconfiguration problem aims at finding a sequence of moves that reconfigures the modules from one given configuration to another. The sliding cube model (in which modules are allowed to slide over the face or edge of neighboring modules) is one of the most studied theoretical models for modular robots. In the sliding cubes model we can reconfigure between any two shapes in $O(n^2)$ moves ([Abel \textit{et al.} SoCG 2024]). If we are interested in a reconfiguration algorithm into a \textit{compact configuration}, the number of moves can be reduced to the sum of coordinates of the input configuration (a number that ranges from $\Omega(n^{4/3})$ to $O(n^2)$, [Kostitsyna \textit{et al.} SWAT 2024]). We introduce a new algorithm that combines both universal reconfiguration and an input-sensitive bound on the sum of coordinates of both configurations, with additional advantages, such as $O(1)$ amortized computation per move.