Natural Computing

, Volume 15, Issue 1, pp 41–49 | Cite as

Producibility in hierarchical self-assembly

Article

Abstract

Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use \(O(|\alpha | \log ^2 |\alpha |)\) time. Cannon et al. (STACS 2013: proceedings of the thirtieth international symposium on theoretical aspects of computer science. pp 172–184, 2013) showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system \({\mathcal {T}}\) can be solved in \(O(|\alpha |^2 |{\mathcal {T}}| + |\alpha | |{\mathcal {T}}|^2)\) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to \(O(|\alpha | |{\mathcal {T}}| \log |{\mathcal {T}}|)\) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently—i.e., if the positions that they share have the same tile type in each assembly—then their union is also producible .

Keywords

Hierarchical Self-assembly Deterministic 

Notes

Acknowledgments

The author is very grateful to Ho-Lin Chen, David Soloveichik, Damien Woods, Matt Patitz, Scott Summers, Robbie Schweller, Ján Maňuch, Ladislav Stacho, Andrew Winslow for many insightful discussions, and to anonymous reviewers for their detailed and useful comments. The author was supported by NSF Grants CCF-1219274 and CCF-1162589 and the Molecular Programming Project under NSF Grants 0832824 and 1317694 and by a Computing Innovation Fellowship under NSF Grant 1019343.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.University of CaliforniaDavisUSA

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