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, Cheng, Goel, and Huang (\emph{Running Time and Program Size for Self-Assembled Squares}, STOC 2001) 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 is 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 \emph{any} shape with diameter $N$ in less than time $\Omega(N)$, demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the $\Omega(N)$ 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.

The previous version of this paper has been split into two papers, and a new result (sublinear time assembly of a shape) has been added to this paper. The previous other half of this paper is here: http://arxiv.org/abs/1011.3493

We settle a number of questions in 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, Cheng, Goel, and Huang ("Running Time and Program Size for Self-Assembled Squares", STOC 2001) showed how to assemble an n x n square in O(n) time in the seeded aTAM using O(log n / log log n) unique tile types, showed that both of these parameters are optimal, and 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 there is a tile system with the optimal O(log n / log log n) tile types that assembles an n x n square using O(log^2 n) parallel "stages", which is close to the optimal Omega(log n) stages, forming the final n x n from four n/2 x n/2 squares, which are themselves recursively formed from n/4 x n/4 squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We leave open the question of whether some hierarchical tile system can break the Omega(n) assembly time lower bound for assembling an n x n 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 N in less than time Omega(N), demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the Omega(N) time lower bound for deterministic seeded systems of Adleman et al. to arbitrary seeded systems.

We then investigate the relationship between the temperature of a tile system and its size. We show that a tile system can in general require temperature that is exponentially greater than its number of tile types. On the other hand, for the special case of *2-cooperative* systems, in which all binding events involve at most 2 sides of tiles, it suffices to use temperature linear in the number of tile types. We show that there is a polynomial-time algorithm that, given any tile system T specified by its desired binding behavior, finds a temperature and binding energies (at most exponential in the number of tile types of T) that realize this behavior or reports that no such energies exist. This result is applied to show that there is a polynomial-time algorithm that, given an n x n square S_n, determines the smallest (non-hierarchical "seeded") system (at any temperature) that is deterministic and self-assembles S_n. This answers an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanes, and Rothemund ("Combinatorial Optimization Problems in Self-Assembly", STOC 2002).