Difference between revisions of "Building n by n squares"
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==Building $n \times n$ squares== | ==Building $n \times n$ squares== | ||
Since Winfree showed in his thesis <ref name=Winf98 /> that the aTAM is computationally universal, we know that we can algorithmically direct the growth of assemblies. This ability allows for not only the creation of complicated and precise shapes, but also often for them to be created very tile type efficiently (i.e. they require small tile sets - those with few numbers of unique tile types). A benchmark problem for tile-based self-assembly is that of assembling an $n \times n$ square since this requires that the tiles somehow compute the value of $n$ and thus "know when to stop" at the boundaries. In <ref name =RotWin00> Rothemund and Winfree showed that binary counters can be used to guide the growth of squares and that thereby it is possible to self-assemble an $n \times n$ square using $O(\log n)$ tile types. | Since Winfree showed in his thesis <ref name=Winf98 /> that the aTAM is computationally universal, we know that we can algorithmically direct the growth of assemblies. This ability allows for not only the creation of complicated and precise shapes, but also often for them to be created very tile type efficiently (i.e. they require small tile sets - those with few numbers of unique tile types). A benchmark problem for tile-based self-assembly is that of assembling an $n \times n$ square since this requires that the tiles somehow compute the value of $n$ and thus "know when to stop" at the boundaries. In <ref name =RotWin00> Rothemund and Winfree showed that binary counters can be used to guide the growth of squares and that thereby it is possible to self-assemble an $n \times n$ square using $O(\log n)$ tile types. | ||
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Revision as of 10:34, 11 June 2013
Contents
Building \(n \times n\) squares
Since Winfree showed in his thesis [1] that the aTAM is computationally universal, we know that we can algorithmically direct the growth of assemblies. This ability allows for not only the creation of complicated and precise shapes, but also often for them to be created very tile type efficiently (i.e. they require small tile sets - those with few numbers of unique tile types). A benchmark problem for tile-based self-assembly is that of assembling an \(n \times n\) square since this requires that the tiles somehow compute the value of \(n\) and thus "know when to stop" at the boundaries. In Cite error: Closing </ref> missing for <ref> tag
</references>
- ↑ Cite error: Invalid
<ref>tag; no text was provided for refs namedWinf98 - ↑ 2.0 2.1
Paul W. K. Rothemund, Erik Winfree - The Program-size Complexity of Self-Assembled Squares (extended abstract)
- STOC '00: Proceedings of the thirty-second annual ACM Symposium on Theory of Computing pp. 459--468, Portland, Oregon, United States,2000
- BibtexAuthor : Paul W. K. Rothemund, Erik Winfree
Title : The Program-size Complexity of Self-Assembled Squares (extended abstract)
In : STOC '00: Proceedings of the thirty-second annual ACM Symposium on Theory of Computing -
Address : Portland, Oregon, United States
Date : 2000
- ↑
Paul W. K. Rothemund - Theory and Experiments in Algorithmic Self-Assembly
- Ph.D. Thesis, University of Southern California , December 2001
- BibtexAuthor : Paul W. K. Rothemund
Title : Theory and Experiments in Algorithmic Self-Assembly
In : Ph.D. Thesis, University of Southern California -
Address :
Date : December 2001
- ↑
James I. Lathrop, Jack H. Lutz, Scott M. Summers - Strict Self-Assembly of Discrete Sierpinski Triangles
- Theoretical Computer Science 410:384--405,2009
- BibtexAuthor : James I. Lathrop, Jack H. Lutz, Scott M. Summers
Title : Strict Self-Assembly of Discrete Sierpinski Triangles
In : Theoretical Computer Science -
Address :
Date : 2009
- ↑
David Soloveichik, Erik Winfree - Complexity of Self-Assembled Shapes
- SIAM Journal on Computing 36(6):1544-1569,2007
- BibtexAuthor : David Soloveichik, Erik Winfree
Title : Complexity of Self-Assembled Shapes
In : SIAM Journal on Computing -
Address :
Date : 2007
