## Abstract

In this paper, we extend Rothemund and Winfree's examination of the tile complexity of tile self-assembly. They provided a lower bound of Ω(log N/ log log N) on the tile complexity of assembling an N×N square for almost all N. Adleman et al. gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size O(√log N) which assembles an N×N square in a model which allows flexible glue strength between non-equal glues (This was independently discovered in [3]). This result is matched by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the Ω(log N/ log log N) lower bound applies to N×N squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of Ω(N ^{1/k}/k) for the standard model, yet we also give a construction which achieves O(log N/log logN) complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape, and show that this problem is NP-hard.

Original language | English (US) |
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Pages | 873-882 |

Number of pages | 10 |

State | Published - Apr 15 2004 |

Event | Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms - New Orleans, LA., United States Duration: Jan 11 2004 → Jan 13 2004 |

### Other

Other | Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | New Orleans, LA. |

Period | 1/11/04 → 1/13/04 |

## ASJC Scopus subject areas

- Software
- General Mathematics