Randomized Self-Assembly for Exact Shapes

Abstract: Working in Winfree's abstract tile assembly model, we show that a constant-size tile assembly system can be programmed through relative tile concentrations to build an n x n square with high probability, for any sufficiently large n. This answers an open question of Kao and Schweller (Randomized Self-Assembly for Approximate Shapes, ICALP 2008), who showed how to build an approximately n x n square using tile concentration programming, and asked whether the approximation could be made exact with high probability. We show how this technique can be modified to answer another question of Kao and Schweller, by showing that a constant-size tile assembly system can be programmed through tile concentrations to assemble arbitrary finite scaled shapes, which are shapes modified by replacing each point with a c x c block of points, for some integer c. Furthermore, we exhibit a smooth tradeoff between specifying bits of n via tile concentrations versus specifying them via hard-coded tile types, which allows tile concentration programming to be employed for specifying a fraction of the bits of "input" to a tile assembly system, under the constraint that concentrations can only be specified to a limited precision. Finally, to account for some unrealistic aspects of the tile concentration programming model, we show how to modify the construction to use only concentrations that are arbitrarily close to uniform.