A Limit to the Power of Multiple Nucleation in Self-Assembly (full version)

Majumder, Reif and Sahu presented a model of reversible, error-permitting tile self-assembly, and showed that restricted classes of tile assembly systems achieved equilibrium in (expected) polynomial time. One open question they asked was how the model would change if it permitted multiple nucleation, i.e., independent groups of tiles growing before attaching to the original seed assembly. This paper provides a partial answer, by proving that no tile assembly model can use multiple nucleation to achieve speedup from polynomial time to constant time without sacrificing computational power: if a tile assembly system T uses multiple nucleation to tile a surface in constant time (independent of the size of the surface), then T is unable to solve computational problems that have low complexity in the (single-seeded) Winfree-Rothemund Tile Assembly Model. Moreover, this time bound applies to macroscale robotic systems that assemble in three-space, not just to tile assembly systems on a two-dimensional surface. The proof technique defines a new model of distributed computing that simulates tile (and robotic) self-assembly, so a tile assembly model can be described as a distributed computing model. Keywords: self-assembly, multiple nucleation, locally checkable labeling.