On Efficient Connectivity-Preserving Transformations in a Grid

We consider a discrete system of $n$ devices lying on a 2-dimensional square grid and forming an initial connected shape $S_I$. Each device is equipped with a linear-strength mechanism which enables it to move a whole line of consecutive devices in a single time-step. We study the problem of transforming $S_I$ into a given connected target shape $S_F$ of the same number of devices, via a finite sequence of \emph{line moves}. Our focus is on designing \emph{centralised} transformations aiming at \emph{minimising the total number of moves} subject to the constraint of \emph{preserving connectivity} of the shape throughout the course of the transformation. We first give very fast connectivity-preserving transformations for the case in which the \emph{associated graphs} of $S_I$ and $S_F$ are isomorphic to a Hamiltonian line. In particular, our transformations make $O(n \log n$) moves, which is asymptotically equal to the best known running time of connectivity-breaking transformations. Our most general result is then a connectivity-preserving \emph{universal transformation} that can transform any initial connected shape $S_I$ into any target connected shape $S_F$, through a sequence of $O(n\sqrt{n})$ moves. Finally, we establish $\Omega(n \log n)$ lower bounds for two restricted sets of transformations. These are the first lower bounds for this model and are matching the best known $O(n \log n)$ upper bounds.