# Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability

@inproceedings{Fekete2015SizeDependentTS,
title={Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability},
author={S{\'a}ndor P. Fekete and Robert T. Schweller and Andrew Winslow},
booktitle={ISAAC},
year={2015}
}
• Published in ISAAC 23 September 2015
• Computer Science
We introduce a new model of algorithmic tile self-assembly called size-dependent assembly. In previous models, supertiles are stable when the total strength of the bonds between any two halves exceeds some constant temperature. In this model, this constant temperature requirement is replaced by an nondecreasing temperature function$$\tau : \mathbb {N} \rightarrow \mathbb {N}$$ that depends on the size of the smaller of the two halves. This generalization allows supertiles to become unstable and…
1 Citations

## References

SHOWING 1-10 OF 22 REFERENCES

### Self-assemblying Classes of Shapes with a Minimum Number of Tiles, and in Optimal Time

• Mathematics, Computer Science
FSTTCS
• 2006
This paper constructs fixed finite tile systems that assemble into particular classes of shapes that are optimal for rectangles and squares and introduces the notion of parallel time, which is a good approximation of the classical asynchronous time.

### The program-size complexity of self-assembled squares (extended abstract)

• Computer Science
STOC '00
• 2000
A formal model of pseudocrystalline self-assembly, called the Tile Assembly Model, in which a tile may be added to the growing object when the total interaction strength with its neighbors exceeds a parameter Τ is studied, which finds a dramatic decrease in complexity.

### Reducing Tile Complexity for the Self-assembly of Scaled Shapes Through Temperature Programming

There is no constant-size tile set that can uniquely assemble an arbitrary (non-scaled, connected) shape in the multiple temperature model, i.e., the scaling is necessary for self-assembly.

### Randomized Self-Assembly for Exact Shapes

• David Doty
• Mathematics
2009 50th Annual IEEE Symposium on Foundations of Computer Science
• 2009
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

### Identifying Shapes Using Self-assembly

• Mathematics
Algorithmica
• 2011
In this paper, we introduce the following problem in the theory of algorithmic self-assembly: given an input shape as the seed of a tile-based self-assembly system, design a finite tile set that can,

### Reducing tile complexity for self-assembly through temperature programming

• Computer Science
SODA '06
• 2006
This work suggests that temperature change can constitute a natural, dynamic method for providing input to self-assembly systems that is potentially superior to the current technique of designing large tile sets with specific inputs hardwired into the tileset.

### Negative Interactions in Irreversible Self-assembly

• Computer Science
Algorithmica
• 2012
The power of negative interactions with irreversible attachments is investigated, and an impossibility theorem is achieved: after t steps of assembly, Ω(t) tiles will be forever bound to an assembly, unable to detach, and negative glue strengths do not afford unlimited power to reuse tiles.

### Randomized Self-assembly for Approximate Shapes

• Materials Science
ICALP
• 2008
T tile self-assembly systems which assemble arbitrarily close approximations to target squares with arbitrarily high probability are designed, in contrast to previous work which has only considered deterministic assemblies of a single shape.

### Complexities for generalized models of self-assembly

• Computer Science
SODA '04
• 2004
This paper considers whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model, and investigates the problem of verifying whether a given tile system uniquely assembles into a given shape, and shows that this problem is NP-hard.

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

• Computer Science
STACS
• 2013
This work shows how to simulate any seeded system with a two- handed system that is essentially just a constant factor larger, and shows 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.