We consider staged self-assembly systems, in which square-shaped tiles can beadded to bins in several stages. Within these bins, the tiles may connect toeach other, depending on the glue types of their edges. Previous work byDemaine et al. showed that a relatively small number of tile types suffices toproduce arbitrary shapes in this model. However, these constructions were onlybased on a spanning tree of the geometric shape, so they did not produce fullconnectivity of the underlying grid graph in the case of shapes with holes;designing fully connected assemblies with a polylogarithmic number of stageswas left as a major open problem. We resolve this challenge by presenting newsystems for staged assembly that produce fully connected polyominoes in O(log^2n) stages, for various scale factors and temperature {\tau} = 2 as well as{\tau} = 1. Our constructions work even for shapes with holes and uses only aconstant number of glues and tiles. Moreover, the underlying approach is moregeometric in nature, implying that it promised to be more feasible for shapeswith compact geometric description.
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