We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{ O}(\frac{ \log{ n} - tb - t\log t}{ b^2} + \frac{ \log \log b}{ \log t})$ stages suffice and $\Omega(\frac{ \log{ n} - tb - t\log t}{ b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{ O}(\frac{ K(S) - tb - t\log t}{ b^2} + \frac{ \log \log b}{ \log t})$ stages suffice and $\Omega(\frac{ K(S) - tb - t\log t}{ b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.
Categories: cs.CG cs.ET