Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The $O(1)$ Musketeers

TL;DR

This paper introduces a universal reconfiguration algorithm for facet-connected modular robots using pivot moves, requiring only five extra modules and achieving worst-case optimal move complexity.

Contribution

It presents the first universal reconfiguration method for any two configurations with a small constant number of helper modules, improving over previous restrictive approaches.

Findings

Five helper modules suffice for universal reconfiguration.

The algorithm uses $O(n^2)$ pivot moves, which is worst-case optimal.

Configurations with certain local patterns can have disconnected reconfiguration graphs.

Abstract

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra "helper" modules ("musketeers") suffice to reconfigure the remaining $n$ modules between any two given configurations. Our algorithm uses $O(n^2)$ pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive "sliding" moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.