Producibility in hierarchical self-assembly

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 ? is producible. The algorithm can be optimized to use $$O(|\alpha | \log ^2 |\alpha |)$$O(|?|log2|?|) 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 ? is the unique producible terminal assembly of a tile system $${\mathcal {T}}$$T can be solved in $$O(|\alpha |^2 |{\mathcal {T}}| + |\alpha | |{\mathcal {T}}|^2)$$O(|?|2|T|+|?||T|2) time for the special case of noncooperative temperature 1 systems. It is shown that this can be improved to $$O(|\alpha | |{\mathcal {T}}| \log |{\mathcal {T}}|)$$O(|?||T|log|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 .