Parallel bounds for cube sliding and leaf-focused sliding-by-refolding models

Establish whether there exist parallel reconfiguration algorithms in the cube sliding model or the leaf-focused sliding-by-refolding model that achieve similarly strong bounds to those known in other modular-robotics models, such as moving a linear number of cubes per round.

Background

Sequential algorithms for reconfiguring tree-shaped polycubes via sliding and rotating leaf cubes take O(n^2) steps. In contrast, other modular robotics models admit parallel algorithms moving a linear number of modules per round.

The authors explicitly state that achieving analogous parallel efficiency remains an open problem in the cube sliding and leaf-focused refolding settings.