We propose and study a class of multi-object rearrangement problems in which a robotic manipulator, capable of carrying an item and making item swaps, is tasked to sort items stored in lattices in a time-optimal manner. We systematically analyze the intrinsic optimality structure, which is fairly rich and intriguing, under different levels of item distinguishability (fully labeled, where each item has a unique label, or partially labeled, where multiple items may be of the same type) and different lattice dimensions. Focusing on the most practical setting of one and two dimensions, we develop efficient (low polynomial time) algorithms that optimally perform rearrangements on 1D lattices under both fully- and partially-labeled settings. On the other hand, we prove that rearrangement on 2D and higher dimensional lattices becomes computationally intractable to optimally solve. Despite their NP-hardness, we are able to again develop efficient algorithms for 2D fully- and partially-labeled settings that are asymptotically optimal, in expectation, assuming that the initial configuration is randomly selected. Simulation studies confirm the effectiveness of our algorithms in comparison to greedy best-first approaches. Source code of Python implementation: https://github.com/rutgers-arc-lab/lattice-rearrangement/
