Parallelism and Time in Hierarchical Self-Assembly

Abstract

We study the role that parallelism plays in time complexity of variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the “hierarchical” aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the “seeded” aTAM, tiles attach one at a time to a growing assembly. Adleman et al. [Running time and program size for self-assembled squares, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (Hersonissos, Greece), ACM, New York, 2001, pp. 740--748] showed how to assemble an $n \times n$ square in $O(n)$ time in the seeded aTAM using $O(\frac{\log n}{\log \log n})$ unique tile types, where both of these parameters are optimal. They asked whether the hierarchical aTAM could allow a tile system to use the ability to form large assemblies in parallel before they attach to break the $\Omega(n)$ lower bound for assembly time. We show that there is a tile system with the optimal $O(\frac{\log n}{\log \log n})$ tile types that assembles an $n \times n$ square using $O(\log^2 n)$ parallel “stages,” which are close to the optimal $\Omega(\log n)$ stages, forming the final $n \times n$ square from four $n/2 \times n/2$ squares, which are themselves recursively formed from $n/4 \times n/4$ squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We extend the definition of partial order tile systems studied by Adleman et al. in a natural way to hierarchical assembly and show that no hierarchical partial order tile system can build any shape with diameter $D$ in less than time $\Omega(D)$, demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the $\Omega(D)$ time lower bound for deterministic seeded systems of Adleman et al. to nondeterministic seeded systems. Finally, we show that for infinitely many $n$, a tile system can assemble an $n \times n'$ rectangle, with $n > n'$, in time $O(n^{4/5} \log n)$, breaking the linear-time lower bound that applies to all seeded systems and partial order hierarchical systems.