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In the modular robot reconfiguration problem, we are given $n$ cube-shapedmodules (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 sequenceof moves that reconfigures the modules from one configuration to the otherusing "sliding moves," in which a module slides over the face or edge of aneighboring module, maintaining connectivity of the configuration at all times.  For many years it has been known that certain module configurations in thismodel require at least $\Omega(n^2)$ moves to reconfigure between them. In thispaper, we introduce the first universal reconfiguration algorithm -- i.e., weshow that any $n$-module configuration can reconfigure itself into anyspecified $n$-module configuration using just sliding moves. Our algorithmachieves reconfiguration in $O(n^2)$ moves, making it asymptotically tight. Wealso present a variation that reconfigures in-place, it ensures that throughoutthe reconfiguration process, all modules, except for one, will be contained inthe 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