Rubik Tables and object rearrangement

Abstract

A great number of robotics applications demand the rearrangement of many mobile objects, for example, organizing products on store shelves, shuffling containers at shipping ports, reconfiguring fleets of mobile robots, and so on. To boost the efficiency/throughput in systems designed for solving these rearrangement problems, it is essential to minimize the number of atomic operations that are involved, for example, the pick-n-places of individual objects. However, this optimization task poses a rather difficult challenge due to the complex inter-dependency between the objects, especially when they are tightly packed together. In this work, in tackling the aforementioned challenges, we have developed a novel algorithmic tool, called Rubik Tables, that provides a clean abstraction of object rearrangement problems as the proxy problem of shuffling items stored in a table or lattice. In its basic form, a Rubik Table is an n × n table containing n2 items. We show that the reconfiguration of items in such a Rubik Table can be achieved using at most n column and n row shuffles in the partially labeled setting, where each column (resp., row) shuffle may arbitrarily permute the items stored in a column (resp., row) of the table. When items are fully distinguishable, additional n shuffles are needed. Rubik Tables allow many generalizations, for example, adding an additional depth dimension or extending to higher dimensions. Using Rubik Table results, we have designed a first constant-factor optimal algorithm for stack rearrangement problems where items are stored in stacks, accessible only from the top. We show that, for nd items stored in n stacks of depth d each, using one empty stack as the swap space, O(nd) stack pop-push operations are sufficient for an arbitrary reconfiguration of the stacks where for arbitrary fixed m > 0. Rubik Table results also allow the development of constant-factor optimal solutions for solving multi-robot motion planning problems under extreme robot density. These algorithms based on Rubik Table results run in low-polynomial time.