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 α is producible. The algorithm can be optimized to use O(|α| log2 |α|) time. Cannon, Demaine, Demaine, Eisenstat, Patitz, Schweller, Summers, and Winslow [4] 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.