The sliding cubes model serves as a well-established theoretical framework for formalizing and analyzing reconfiguration algorithms in modular robotic systems built from face-connected cubic modules. We extend the parallel sliding cubes model from two to three dimensions, presenting new algorithms, surprising complexity results, and a generalization of the best known bounds from two to three dimensions. A companion video visualizes and explains our results.

In the classic sliding cube model for programmable matter in three dimensions, the task is to find a reconfiguration sequence between two connected configurations of n indistinguishable unit cube modules by sliding modules along their neighbors' faces. Depending on the objective, this sequence should minimize either the total energy expended (the number of moves) or the total elapsed time (the makespan). We give a number of results for the three-dimensional setting, including (i) the first algorithm that achieves worst-case optimal makespan under parallel motion in three dimensions, (ii) a proof of log-APX-hardness to decide either the optimal makespan or the optimal number of moves, which is the strongest known inapproximability bound in any related model, and (iii) a proof of NP-hardness to decide the optimal makespan under parallel motion, even if the two configurations differ only by one module and the optimal makespan is at most two. Our results strengthen the inapproximability claim from [Hugo A. Akitaya et al., 2022] and answer a question of [Akitaya et al., 2025] in the negative.