Before revision:  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.
After revision:  Focusing on the most practical setting of one and two dimensions, we develop efficient ( low polynomial time cycle-following based algorithms that optimally perform rearrangements on 1D lattices under both fully and partially labeled settings.