Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types t , bins b , and stages necessary to assemble n × n squares and scaled shapes in the staged tile assembly model. For n × n squares, we prove 𝒪( logn - tb - tlog t/b^2 + loglog b/log t) stages suffice and ( logn - tb - tlog t/b^2) are necessary for almost all n . For shapes S with Kolmogorov complexity K ( S ), we prove 𝒪( K(S) - tb - tlog t/b^2 + loglog b/log t) stages suffice and ( K(S) - tb - tlog 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.