Andrew Winslow

Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM

We study the dierence between the standard seeded model (aTAM) of tile self-assembly, and the “seedless” two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model. 1998 ACM Subject Classification F.1.2

One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile

In the classical model of tile self-assembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of self-assembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via information-theoretic arguments, increasingly complex shapes necessarily require increasing numbers of distinct types of tiles.

One-dimensional staged self-assembly

We introduce the problem of staged self-assembly of one-dimensional nanostructures, which becomes interesting when the elements are labeled (e.g., representing functional units that must be placed at specific locations). In a restricted model in which each operation has a single terminal assembly, we prove that assembling a given string of labels with the fewest stages is equivalent, up to constant factors, to compressing the string to be uniquely derived from the smallest possible context-free grammar (a well-studied O(log n)-approximable problem). Without this restriction, we show that the optimal assembly can be substantially smaller than the optimal context-free grammar, by a factor of Ω(√n/ log n) even for binary strings of length n. Fortunately, we can bound this separation in model power by a quadratic function in the number of distinct glues or tiles allowed in the assembly, which is typically small in practice.

Staged self-assembly and polyomino context-free grammars

Previous work by Demaine et al. (Nat Comput 6937:100–114, 2012) developed a strong connection between smallest context-free grammars and staged self-assembly systems for one-dimensional strings and assemblies. We extend this work to two-dimensional polyominoes and assemblies, comparing staged self-assembly systems to a natural generalization of context-free grammars we call polyomino context-free grammars (PCFGs). We achieve nearly optimal bounds on the largest ratios of the smallest PCFG and staged self-assembly system for a given polyomino with

Tight Bounds for Active Self-assembly Using an Insertion Primitive

n

Optimal Staged Self-Assembly of General Shapes

n cells. For the ratio of PCFGs over assembly systems, we show that the smallest PCFG can be an

A Brief Tour of Theoretical Tile Self-Assembly

\varOmega (n/\log ^3{n})