Randomized Self-Assembly for Exact Shapes

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.

💡 Research Summary

This paper studies the abstract Tile Assembly Model (aTAM) introduced by Winfree, focusing on the “tile concentration programming” input paradigm. In this model, a fixed set of tile types is supplied in solution at specified concentrations; when a frontier location is ready to accept a tile, the probability of each tile type binding is proportional to its concentration. The authors address two open questions posed by Kao and Schweller (ICALP 2008): (1) whether a constant‑size tile set can be programmed via concentrations to assemble an exact n × n square with high probability, and (2) whether arbitrary finite shapes, possibly scaled by a factor c, can be assembled in the same way.

The main contributions are as follows:

1. Exact n × n Square Construction
   The authors design a constant‑size tile set that, for any sufficiently large integer n, can be programmed by setting relative concentrations to produce an exact n × n square with probability at least 1 − δ (for any prescribed δ > 0). The construction proceeds in two stages. First, a thin rectangular “sampling structure” of height O(log n) and length O(n^{2/3}) (or O(n^{ε}) for any ε > 0) self‑assembles. This rectangle encodes the binary representation of n: each bit is represented by a pair of tile types (0‑tile and 1‑tile) whose relative concentrations are set to bias the stochastic attachment toward the correct bit. By applying Chernoff bounds to the independent attachment events, the authors guarantee that the entire binary string is formed correctly with probability 1 − δ.

   Second, the binary string is interpreted to control the growth of the rectangle’s width and height, effectively “reading” the value n and extending the assembly to exactly n columns and n rows. The authors use a length‑conversion sub‑assembly that replicates each bit many times across the rectangle, allowing a deterministic‑looking growth phase despite the underlying randomness. Because aTAM is monotone (tiles never detach), any erroneous attachment cannot later destroy the structure; the probabilistic analysis ensures that such errors are exceedingly unlikely.

2. Arbitrary Scaled Shapes
   Building on the square construction, the paper shows how to assemble any finite connected shape S after scaling each point of S to a c × c block (for some integer c that may depend on S). The binary encoding of n is replaced by a “seed block” that enumerates the points of S in a suitable order. By feeding this seed block into the same growth machinery used for the square, the assembly produces a scaled copy of S. This answers the second question of Kao and Schweller affirmatively.

3. Trade‑off Between Concentration Bits and Tile Types
   Encoding log n bits of information solely via concentrations would require arbitrarily fine concentration precision, which is unrealistic. The authors therefore present a smooth trade‑off: let g ≤ log n bits be encoded by concentrations, and the remaining log n − g bits be hard‑coded into the tile set (i.e., by using 2^{g} distinct tile types). Theorem 5.3 quantifies the exact relationship, showing that the total description length (bits needed to specify concentrations plus bits needed to describe the tile set) remains Θ(log n). By choosing g appropriately, one can balance concentration precision against the number of distinct tile types, achieving asymptotically optimal resource usage.

4. Near‑Uniform Concentrations and Kinetic Realism
   The original concentration programming model assumes arbitrary concentration ratios, which is not physically realistic. Moreover, Winfree’s proof that aTAM can be approximated by the kinetic Tile Assembly Model (kTAM) relies on uniform concentrations. To reconcile this, the authors modify their construction so that only concentrations arbitrarily close to uniform (all ρ(t)∈