Parallelism and time in hierarchical self-assembly

We study the role that parallelism plays in time complexariants 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×n square in O(n) time in the seeded aTAM using O( 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 Ω(n) lower bound for assembly time. We show that there is a tile system with the optimal O( log n log log n ) tile types that assembles an n×n square using O(log2 n) parallel “stages”, which is close to the op...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Ho-Lin Chen, David Doty
• Keywords: Algorithms, Maximal Parallelism, SODA 2012, Tile Systems, Tile Types