11,875 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
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
Programmable Control of Nucleation for Algorithmic Self-Assembly
Algorithmic self-assembly, a generalization of crystal growth processes, has
been proposed as a mechanism for autonomous DNA computation and for bottom-up
fabrication of complex nanostructures. A `program' for growing a desired
structure consists of a set of molecular `tiles' designed to have specific
binding interactions. A key challenge to making algorithmic self-assembly
practical is designing tile set programs that make assembly robust to errors
that occur during initiation and growth. One method for the controlled
initiation of assembly, often seen in biology, is the use of a seed or catalyst
molecule that reduces an otherwise large kinetic barrier to nucleation. Here we
show how to program algorithmic self-assembly similarly, such that seeded
assembly proceeds quickly but there is an arbitrarily large kinetic barrier to
unseeded growth. We demonstrate this technique by introducing a family of tile
sets for which we rigorously prove that, under the right physical conditions,
linearly increasing the size of the tile set exponentially reduces the rate of
spurious nucleation. Simulations of these `zig-zag' tile sets suggest that
under plausible experimental conditions, it is possible to grow large seeded
crystals in just a few hours such that less than 1 percent of crystals are
spuriously nucleated. Simulation results also suggest that zig-zag tile sets
could be used for detection of single DNA strands. Together with prior work
showing that tile sets can be made robust to errors during properly initiated
growth, this work demonstrates that growth of objects via algorithmic
self-assembly can proceed both efficiently and with an arbitrarily low error
rate, even in a model where local growth rules are probabilistic.Comment: 37 pages, 14 figure
Verification in Staged Tile Self-Assembly
We prove the unique assembly and unique shape verification problems,
benchmark measures of self-assembly model power, are
-hard and contained in (and in
for staged systems with stages). En route,
we prove that unique shape verification problem in the 2HAM is
-complete.Comment: An abstract version will appear in the proceedings of UCNC 201
Reflections on Tiles (in Self-Assembly)
We define the Reflexive Tile Assembly Model (RTAM), which is obtained from
the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across
their horizontal and/or vertical axes. We show that the class of directed
temperature-1 RTAM systems is not computationally universal, which is
conjectured but unproven for the aTAM, and like the aTAM, the RTAM is
computationally universal at temperature 2. We then show that at temperature 1,
when starting from a single tile seed, the RTAM is capable of assembling n x n
squares for n odd using only n tile types, but incapable of assembling n x n
squares for n even. Moreover, we show that n is a lower bound on the number of
tile types needed to assemble n x n squares for n odd in the temperature-1
RTAM. The conjectured lower bound for temperature-1 aTAM systems is 2n-1.
Finally, we give preliminary results toward the classification of which finite
connected shapes in Z^2 can be assembled (strictly or weakly) by a singly
seeded (i.e. seed of size 1) RTAM system, including a complete classification
of which finite connected shapes be strictly assembled by a "mismatch-free"
singly seeded RTAM system.Comment: New results which classify the types of shapes which can
self-assemble in the RTAM have been adde
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