We study the classic sliding cube model for programmable matter under parallel reconfiguration in three dimensions, providing novel algorithmic and surprising complexity results in addition to generalizing the best known bounds from two to three dimensions. In general, the problem asks for reconfiguration sequences between two connected configurations of $n$ indistinguishable unit cube modules under connectivity constraints; a connected backbone must exist at all times. The makespan of a reconfiguration sequence is the number of parallel moves performed. We show that deciding the existence of such a sequence is NP-hard, even for constant makespan and if the two input configurations have constant-size symmetric difference, solving an open question in [Akitaya et al., ESA 25]. In particular, deciding whether the optimal makespan is 1 or 2 is NP-hard. We also show log-APX-hardness of the problem in sequential and parallel models, strengthening the APX-hardness claim in [Akitaya et al., SWAT 22]. Finally, we outline an asymptotically worst-case optimal input-sensitive algorithm for reconfiguration. The produced sequence has length that depends on the bounding box of the input configurations which, in the worst case, results in a $O(n)$ makespan.

翻译：我们研究了三维空间中可编程物质在并行重构下的经典滑动立方体模型，不仅将已知的最佳边界从二维推广至三维，还提出了新颖的算法并揭示了出人意料的复杂度结论。该问题通常要求在两个由$n$个不可区分的单位立方体模块构成的连通构型之间，在保持连通性约束的条件下寻找重构序列；即必须始终存在连通的骨干结构。重构序列的完工时间指所执行的并行移动次数。我们证明，即使完工时间为常数且两个输入构型具有常数规模的对称差，判定此类序列的存在性仍是NP难问题，这解决了[Akitaya等人，ESA 25]中提出的开放性问题。特别地，判定最优完工时间为1还是2是NP难的。我们还证明了该问题在串行与并行模型中的对数APX难度，强化了[Akitaya等人，SWAT 22]中关于APX难度的论断。最后，我们提出了一种对输入敏感且渐近最坏情况最优的重构算法。所生成序列的长度取决于输入构型的边界框尺寸，在最坏情况下会产生$O(n)$的完工时间。