New Geometric Algorithms for Staged Self-Assembly

We consider staged self-assembly, in which squareshaped Wang tiles can be added to bins in several stages. Within these bins the tiles may connect to each other, depending on the glue types of their edges. In general, self-assembly constructs complex (polyomino-shaped) structures from a limited set of square tiles. Previous work by Demaine et al. considered a setting in which assembly proceeds in stages. It was shown that a relatively small number of tile types suffices to produce arbitrary shapes; however, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph. We present new systems for stages assembly to assemble a fully connected polyomino in O(log n) stages. Our construction works even for shapes with holes and uses only a constant number of glues and tiles.