Speed of assembly

An important efficiency measure is tile complexity. However, another interesting and important measure which has been investigated is the speed at which an assembly can form, or the number of assembly steps required by a system to reach the final, desired target structure. Of course, in the basic aTAM where each step of assembly consists of a single tile addition, the assembly time for a shape consisting of n points cannot vary, and is fixed at n - s steps (where s is the number of tiles in the seed, usually 1). However, by considering slight variants of the model, such as a version where at each time step all tiles which are able to individually attach do so, the assembly time becomes variable and an interesting metric.

In , Adleman, Cheng, Goel, and Huang proved that the deterministic assembly of a shape of diameter d requires time \(\Omega(d)\). In Doty and Chen showed that this bound also holds for nondeterministic systems. In they provided a matching upper bound for a construction which was able to self-assemble an n × n square and also used the optimal \(O(\frac{\log n}{\log \log n})\) tile types. See Building n by n squares for more discussion of assembly time related to n × n squares.