Producibility in hierarchical self-assembly

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(|α|log2|α|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|\alpha | \log ^2 |\alpha |)$$\end{document} 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 T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}$$\end{document} can be solved in O(|α|2|T|+|α||T|2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|\alpha |^2 |{\mathcal {T}}| + |\alpha | |{\mathcal {T}}|^2)$$\end{document} time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to O(|α||T|log|T|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|\alpha | |{\mathcal {T}}| \log |{\mathcal {T}}|)$$\end{document} 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 .