25,951 research outputs found
Optimal Staged Self-Assembly of General Shapes
We analyze the number of tile types , bins , and stages necessary to
assemble squares and scaled shapes in the staged tile assembly
model. For squares, we prove stages suffice and
are necessary for almost all .
For shapes with Kolmogorov complexity , we prove
stages suffice and are necessary to
assemble a scaled version of , for almost all . We obtain similarly tight
bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201
Intrinsic universality and the computational power of self-assembly
This short survey of recent work in tile self-assembly discusses the use of
simulation to classify and separate the computational and expressive power of
self-assembly models. The journey begins with the result that there is a single
universal tile set that, with proper initialization and scaling, simulates any
tile assembly system. This universal tile set exhibits something stronger than
Turing universality: it captures the geometry and dynamics of any simulated
system. From there we find that there is no such tile set in the
noncooperative, or temperature 1, model, proving it weaker than the full tile
assembly model. In the two-handed or hierarchal model, where large assemblies
can bind together on one step, we encounter an infinite set, of infinite
hierarchies, each with strictly increasing simulation power. Towards the end of
our trip, we find one tile to rule them all: a single rotatable flipable
polygonal tile that can simulate any tile assembly system. It seems this could
be the beginning of a much longer journey, so directions for future work are
suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104
Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability
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 that depends on the size of the smaller of the two halves. This
generalization allows supertiles to become unstable and break apart, and
captures the increased forces that large structures may place on the bonds
holding them together.
We demonstrate the power of this model in two ways. First, we give fixed tile
sets that assemble constant-height rectangles and squares of arbitrary input
size given an appropriate temperature function. Second, we prove that deciding
whether a supertile is stable is coNP-complete. Both results contrast with
known results for fixed temperature.Comment: In proceedings of ISAAC 201
New Geometric Algorithms for Fully Connected Staged Self-Assembly
We consider staged self-assembly systems, in which square-shaped tiles can be
added to bins in several stages. Within these bins, the tiles may connect to
each other, depending on the glue types of their edges. Previous work by
Demaine et al. showed that a relatively small number of tile types suffices to
produce arbitrary shapes in this model. However, these constructions were only
based on a spanning tree of the geometric shape, so they did not produce full
connectivity of the underlying grid graph in the case of shapes with holes;
designing fully connected assemblies with a polylogarithmic number of stages
was left as a major open problem. We resolve this challenge by presenting new
systems for staged assembly that produce fully connected polyominoes in O(log^2
n) stages, for various scale factors and temperature {\tau} = 2 as well as
{\tau} = 1. Our constructions work even for shapes with holes and uses only a
constant number of glues and tiles. Moreover, the underlying approach is more
geometric in nature, implying that it promised to be more feasible for shapes
with compact geometric description.Comment: 21 pages, 14 figures; full version of conference paper in DNA2
Self-Assembly of Arbitrary Shapes Using RNAse Enzymes: Meeting the Kolmogorov Bound with Small Scale Factor (extended abstract)
We consider a model of algorithmic self-assembly of geometric shapes out of
square Wang tiles studied in SODA 2010, in which there are two types of tiles
(e.g., constructed out of DNA and RNA material) and one operation that destroys
all tiles of a particular type (e.g., an RNAse enzyme destroys all RNA tiles).
We show that a single use of this destruction operation enables much more
efficient construction of arbitrary shapes. In particular, an arbitrary shape
can be constructed using an asymptotically optimal number of distinct tile
types (related to the shape's Kolmogorov complexity), after scaling the shape
by only a logarithmic factor. By contrast, without the destruction operation,
the best such result has a scale factor at least linear in the size of the
shape, and is connected only by a spanning tree of the scaled tiles. We also
characterize a large collection of shapes that can be constructed efficiently
without any scaling
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