On the Utility of Buffers in Pick-n-Swap Based Lattice Rearrangement

We investigate the utility of employing multiple buffers in solving a class of rearrangement problems with pick-n-swap manipulation primitives. In this problem, objects stored randomly in a lattice are to be sorted using a robot arm with k>=1 swap spaces or buffers, capable of holding up to k objects on its end-effector simultaneously. On the structural side, we show that the addition of each new buffer brings diminishing returns in saving the end-effector travel distance while holding the total number of pick-n-swap operations at the minimum. This is due to an interesting recursive cycle structure in random m-permutation, where the largest cycle covers over 60% of objects. On the algorithmic side, we propose fast algorithms for 1D and 2D lattice rearrangement problems that can effectively use multiple buffers to boost solution optimality. Numerical experiments demonstrate the efficiency and scalability of our methods, as well as confirm the diminishing return structure as more buffers are employed.