Pushing Lines Helps: Efficient Universal Centralised Transformations for Programmable Matter

Abstract

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 log 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 log n)-time universal transformation. We leave as an interesting open problem a suspected Ω(n log n)-time lower bound. A. Almethen, O. Michail and I. Potapov XX:3such transformations based on pipelining [23,30], where essentially the shape transforms by moving nodes in parallel around its perimeter, can be shown to require O(n) parallel time in the worst case and this technique has also been applied in systems (e.g., [36]).The other approach is to consider more powerful actuation mechanisms, that have the potential to reduce the inherent distance faster than a constant per sequential time-step. These are typically mechanisms where the local actuation has strength higher than a constant. This is different than the above parallel-time transformations, in which local actuation can only move a single node one position in its local neighbourhood and the combined effect of many such movements at the same time is exploited. In contrast, in higher-strength mechanisms, it is a single actuation that has enough strength to move many nodes at the same time. Prominent examples in the literature are the linear-strength models of Aloupis et al. [1,2], in which nodes are equipped with extend/contract arms, each having the strength to extend/contract the whole shape as a result of applying such an operation to one of its neighbours and of Woods et al. [40], in which a whole line of nodes can rotate around a single node (acting as a linear-strength rotating arm). The present paper follows this approach, by intro...