11,875 research outputs found

    Optimal Staged Self-Assembly of General Shapes

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    We analyze the number of tile types tt, bins bb, and stages necessary to assemble n×nn \times n squares and scaled shapes in the staged tile assembly model. For n×nn \times n squares, we prove O(logntbtlogtb2+loglogblogt)\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}) stages suffice and Ω(logntbtlogtb2)\Omega(\frac{\log{n} - tb - t\log t}{b^2}) are necessary for almost all nn. For shapes SS with Kolmogorov complexity K(S)K(S), we prove O(K(S)tbtlogtb2+loglogblogt)\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}) stages suffice and Ω(K(S)tbtlogtb2)\Omega(\frac{K(S) - tb - t\log t}{b^2}) are necessary to assemble a scaled version of SS, for almost all SS. 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

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    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 τ:NN\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 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

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    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

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    We prove the unique assembly and unique shape verification problems, benchmark measures of self-assembly model power, are coNPNP\mathrm{coNP}^{\mathrm{NP}}-hard and contained in PSPACE\mathrm{PSPACE} (and in Π2sP\mathrm{\Pi}^\mathrm{P}_{2s} for staged systems with ss stages). En route, we prove that unique shape verification problem in the 2HAM is coNPNP\mathrm{coNP}^{\mathrm{NP}}-complete.Comment: An abstract version will appear in the proceedings of UCNC 201

    Reflections on Tiles (in Self-Assembly)

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    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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