Step-assembly with a constant number of tile types

Abstract

Demaine et. al. [1] have introduced a model for staged assembly for constructing self-assembled shapes as an extension to the multiple tile model [2]. In their model the assembly proceeds in several bins and several stages with different sets of tiles and supertiles applied on each bin and in each stage. Taking advantage of all these features they showed that a constant number of tile types is sufficient to self-assemble any given shape. In this paper, we consider a simplified model of staged assembly, called the step assembly model, in which we only have one bin in each step and assembly happens by attaching tiles one by one to the growing structure as in the standard assembly model. We show that in this simplified model a constant number of tile types (24) is sufficient to assemble a large class of shapes. This class includes all shapes obtained from any shape by scaling by a factor of 2. For general shapes, we note that the tile complexity of this model has connections to the monotone connected node search number of a spanning tree of the shape. © 2009 Springer-Verlag Berlin Heidelberg.