Producibility in hierarchical self-assembly

TL;DR

This paper presents new algorithms and complexity results for determining producibility and uniqueness in hierarchical tile self-assembly, including efficient decision procedures and properties of assembly unions.

Contribution

It introduces optimized algorithms for deciding producibility and uniqueness, and proves that union of compatible assemblies is also producible in hierarchical self-assembly.

Findings

Greedy polynomial-time algorithm decides assembly producibility.

Improved time complexity for deciding unique producibility in temperature 1 systems.

Union of two compatible producible assemblies is also producible.

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 A is producible. The algorithm can be optimized to use O(|A| log^2 |A|) time. Cannon, Demaine, Demaine, Eisenstat, Patitz, Schweller, Summers, and Winslow showed that the problem of deciding if an assembly A is the unique producible terminal assembly of a tile system T can be solved in O(|A|^2 |T| + |A| |T|^2) time for the special case of noncooperative "temperature 1" systems. It is shown that this can be improved to O(|A| |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.