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Space-Aware Reconfiguration

We consider the problem of reconfiguring a set of physical objects into a desired target configuration, a typical (sub)task in robotics and automation, arising in product assembly, packaging, stocking store shelves, and more. In this paper we address a variant, which we call space-aware reconfiguration, where the goal is to minimize the physical space needed for the reconfiguration, while obeying constraints on the allowable collision-free motions of the objects. Since for given start and target configurations, reconfiguration may be impossible, we translate the entire target configuration rigidly into a location that admits a valid sequence of moves, where each object moves in turn just once, along a straight line, from its starting to its target location, so that the overall physical space required by the start, all intermediate, and target configurations for all the objects is minimized. We investigate two variants of space-aware reconfiguration for the often examined setting of $n$ unit discs in the plane, depending on whether the discs are distinguishable (labeled) or indistinguishable (unlabeled). For the labeled case, we propose a representation of size $O(n^4)$ of the space