Fast Algorithmic Self-assembly of Simple Shapes Using Random Agitation
We study the power of uncontrolled random molecular movement in a model of self-assembly called the nubots model. The nubots model is an asynchronous nondeterministic cellular automaton augmented with rigid-body movement rules (push/pull, deterministically and programmatically applied to specific monomers) and random agitations (nondeterministically applied to every monomer and direction with equal probability all of the time). Previous work on nubots showed how to build simple shapes such as lines and squares quickly—in expected time that is merely logarithmic of their size. These results crucially make use of the programmable rigid-body movement rule: the ability for a single monomer to push or pull large objects quickly, and only at a time and place of the programmers’ choosing. However, in engineered molecular systems, molecular motion is largely uncontrolled and fundamentally random. This raises the question of whether similar results can be achieved in a more restrictive, and perhaps easier to justify, model where uncontrolled random movements, or agitations, are happening throughout the self-assembly process and are the only form of rigid-body movement. We show that this is indeed the case: we give a polylogarithmic expected time construction for squares using agitation, and a sublinear expected time construction to build a line. Such results are impossible in an agitation-free (and movement-free) setting and thus show the benefits of exploiting uncontrolled random movement.
KeywordsSimple Shape Expected Time Hexagonal Grid Monomer State Agitation Step
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- 3.Dabby, N., Chen, H.-L.: Active self-assembly of simple units using an insertion primitive. In: SODA: Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1526–1536 (January 2012)Google Scholar
- 6.Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: FOCS: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pp. 439–446 (October 2012)Google Scholar
- 9.Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology (June 1998)Google Scholar
- 10.Woods, D.: Intrinsic universality and the computational power of self-assembly. In: MCU: Proceedings of Machines, Computations and Universality, September 9-12. Electronic Proceedings in Theoretical Computer Science, vol. 128, pp. 16–22 (2013)Google Scholar
- 11.Woods, D., Chen, H.-L., Goodfriend, S., Dabby, N., Winfree, E., Yin, P.: Active self-assembly of algorithmic shapes and patterns in polylogarithmic time. In ITCS 2013: Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pp. 353–354. ACM (2013) Full version: arXiv:1301.2626 [cs.DS]