Tag: 2017

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Self-Assembly of Shapes at Constant Scale using Repulsive Forces

Self-Assembly of Shapes at Constant Scale using Repulsive Forces.
Austin Luchsinger, Robert Schweller, and Tim Wylie. In Natural Computing. 2018.

Link: https://link.springer.com/article/10.1007/s11047-018-9707-9

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Self-Assembly of Shapes at Constant Scale using Repulsive Forces.
Austin Luchsinger, Robert Schweller, and Tim Wylie.
In Proc. of the 16th Inter. Conf. on Unconventional Computation and Natural Computation (UCNC’17), 2017.

Abstract:

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Concentration Independent Random Number Generation in Tile Self-Assembly

Title: Concentration Independent Random Number Generation in Tile Self-Assembly
Authors:
In this paper we introduce the robust random number generation problem where the goal is to design an abstract tile assembly system (aTAM system) whose terminal assemblies can be split into n partitions such that a resulting assembly of the system lies within each partition with probability 1/n , regardless of the relative concentration assignment of the tile types in the system. First, we show this is possible for n=2n=2 (a robust fair coin flip ) within the aTAM, and that such systems guarantee a worst case O(1)O(1) space usage. We accompany our primary construction with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility, and also prove some negative results. As an application, we combine our coin-flip system with a result of Chandran, Gopalkrishnan, and Reif to show that for any positive integer n , there exists a O(log⁡n)O(log⁡n) tile system that assembles a constant-width linear assembly of expected length n for any concentration assignment. We then extend our robust fair coin flip result to solve the problem of robust random number generation in the aTAM for all n. Two variants of robust random bit generation solutions are presented: an unbounded space solution and a bounded space solution which incurs a small bias. Further, we consider the harder scenario where tile concentrations change arbitrarily at each assembly step and show that while this is not possible in the aTAM, the problem can be solved by exotic tile assembly models from the literature.

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Optimal Staged Self-Assembly of General Shapes

Title: Optimal Staged Self-Assembly of General Shapes
Authors: Cameron Chalk, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, Tim Wylie
Abstract:
We analyze the number of tile types t, bins b, and stages necessary to assemble \(n \times n\) squares and scaled shapes in the staged tile assembly model. For \(n \times n\) squares, we prove \(\mathcal {O}\left( \frac{\log {n} – tb – t\log t}{b^2} + \frac{\log \log b}{\log t}\right) \) stages suffice and \(\varOmega \left( \frac{\log {n} – tb – t\log t}{b^2}\right) \) are necessary for almost all n. For shapes S with Kolmogorov complexity K(S), we prove \(\mathcal {O}\left( \frac{K(S) – tb – t\log t}{b^2} + \frac{\log \log b}{\log t}\right) \) stages suffice and \(\varOmega \left( \frac{K(S) – tb – t\log t}{b^2}\right) \) are necessary to assemble a scaled version of S, for almost all S. We obtain similarly tight bounds when the more powerful flexible glues are permitted.
Published: Arxiv, European Symposium on Algorithms (ESA’16), Algorithmica

URL: https://link.springer.com/article/10.1007/s00453-017-0318-0

Journal Citation: Algorithmica, 80(4), 1383-1409, 2018.

Bibtex:
@article{CMSVWW:2018:Algorithmica,
author=”Chalk, Cameron and Martinez, Eric and Schweller, Robert and Vega, Luis and Winslow, Andrew and Wylie, Tim”,
title=”Optimal Staged Self-Assembly of General Shapes”,
journal=”Algorithmica”,
year=”2018″,
month=”Apr”,
day=”01″,
volume=”80″,
number=”4″,
pages=”1383–1409″,
issn=”1432-0541″,
doi=”10.1007/s00453-017-0318-0″,
url=”https://doi.org/10.1007/s00453-017-0318-0″
}

Conference Citation: Proceedings of the 24th European Symposium of Algorithms (ESA’16), 57, 26:1–26:17, 2016.