Pushing Lines Helps: Efficient Universal Centralised Transformations for Programmable Matter
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In this paper, we study a discrete system of entities residing on a two-dimensional square grid. Each entity is modelled as a node occupying a distinct cell of the grid. The set of all n nodes forms initially a connected shape A. Entities are equipped with a linear-strength pushing mechanism that can push a whole line of entities, from 1 to n, in parallel in a single time-step. A target connected shape B is also provided and the goal is to transform A into B via a sequence of line movements. Existing models based on local movement of individual nodes, such as rotating or sliding a single node, can be shown to be special cases of the present model, therefore their (inefficient, Θ(n^2)) universal transformations carry over. Our main goal is to investigate whether the parallelism inherent in this new type of movement can be exploited for efficient, i.e., sub-quadratic worst-case, transformations. As a first step towards this, we restrict attention solely to centralised transformations and leave the distributed case as a direction for future research. Our results are positive. By focusing on the apparently hard instance of transforming a diagonal A into a straight line B, we first obtain transformations of time O(n√(n)) without and with preserving the connectivity of the shape throughout the transformation. Then, we further improve by providing two O(n n)-time transformations for this problem. By building upon these ideas, we first manage to develop an O(n√(n))-time universal transformation. Our main result is then an O(n n) -time universal transformation. We leave as an interesting open problem a suspected Ω(n n)-time lower bound.
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