Optimal Staged Self-assembly of Linear Assemblies

  • Cameron Chalk
  • Eric Martinez
  • Robert Schweller
  • Luis Vega
  • Andrew Winslow
  • Tim Wylie
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10867)


We analyze the complexity of building linear assemblies, sets of linear assemblies, and \(\mathcal {O}(1)\)-scale general shapes in the staged tile assembly model. For systems with at most b bins and t tile types, we prove that the minimum number of stages to uniquely assemble a \(1 \times n\) line is \(\varTheta (\log _t{n} + \log _b{\frac{n}{t}} + 1)\). Generalizing to \(\mathcal {O}(1) \times n\) lines, we prove the minimum number of stages is \(\mathcal {O}(\frac{\log {n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})\) and \(\varOmega (\frac{\log {n} - tb - t\log t}{b^2})\).

Next, we consider assembling sets of lines and general shapes using \(t = \mathcal {O}(1)\) tile types. We prove that the minimum number of stages needed to assemble a set of k lines of size at most \(\mathcal {O}(1) \times n\) is \(\mathcal {O}(\frac{k\log n}{b^2}+\frac{k\sqrt{\log n}}{b}+\log \log n)\) and \(\varOmega (\frac{k\log n}{b^2})\). In the case that \(b = \mathcal {O}(\sqrt{k})\), the minimum number of stages is \(\varTheta (\log {n})\). The upper bound in this special case is then used to assemble “hefty” shapes of at least logarithmic edge-length-to-edge-count ratio at \(\mathcal {O}(1)\)-scale using \(\mathcal {O}(\sqrt{k})\) bins and optimal \(\mathcal {O}(\log {n})\) stages.


Tile self-assembly Staged self-assembly DNA computing Biocomputing 


  1. 1.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.D., Wasserman, H.: Linear self-assemblies: equilibria, entropy and convergence rates. In: 6th International Conference on Difference Equations and Applications (2001)Google Scholar
  2. 2.
    Adleman, L.M., Cheng, Q., Goel, A., Huang, M.D.A., Kempe, D., de Espanés, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. SIAM J. Comput. 38(6), 2356–2381 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barish, R.D., Schulman, R., Rothemund, P.W.K., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proc. Natl. Acad. Sci. 106(15), 6054–6059 (2009)CrossRefGoogle Scholar
  4. 4.
    Cannon, S., Demaine, E.D., Demaine, M.L., Eisenstat, S., Patitz, M.J., Schweller, R.T., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors): Self-assembly in the 2HAM vs. aTAM. In: STACS 2013, LIPIcs, vol. 20, pp. 172–184. Schloss Dagstuhl (2013)Google Scholar
  5. 5.
    Chalk, C., Martinez, E., Schweller, R., Vega, L., Winslow, A., Wylie, T.: Optimal staged self-assembly of general shapes. Algorithmica 80, 1383–1409 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chandran, H., Gopalkrishnan, N., Reif, J.: Tile complexity of linear assemblies. SIAM J. Comput. 41(4), 1051–1073 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, H.L., Doty, D.: Parallelism and time in hierarchical self-assembly. In: 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1163–1182. SIAM (2012)CrossRefGoogle Scholar
  8. 8.
    Cheng, Q., Aggarwal, G., Goldwasser, M.H., Kao, M.Y., Schweller, R.T., de Espanés, P.M.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34, 1493–1515 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Demaine, E.D., Demaine, M.L., Fekete, S.P., Ishaque, M., Rafalin, E., Schweller, R.T., Souvaine, D.L.: Staged self-assembly: nanomanufacture of arbitrary shapes with \({O}(1)\) glues. Nat. Comput. 7(3), 347–370 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Demaine, E.D., Eisenstat, S., Ishaque, M., Winslow, A.: One-dimensional staged self-assembly. Nat. Comput. 12(2), 247–258 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Demaine, E.D., Fekete, S.P., Scheffer, C., Schmidt, A.: New geometric algorithms for fully connected staged self-assembly. In: Phillips, A., Yin, P. (eds.) DNA 2015. LNCS, vol. 9211, pp. 104–116. Springer, Cham (2015). Scholar
  12. 12.
    Evans, C.: Crystals that count! Physical principles and experimental investigations of DNA tile self-assembly. Ph.D. thesis, Caltech (2014)Google Scholar
  13. 13.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the 32nd ACM Symposium on Theory of Computing, STOC 2000, pp. 459–468 (2000)Google Scholar
  14. 14.
    Schulman, R., Winfree, E.: Synthesis of crystals with a programmable kinetic barrier to nucleation. Proc. Natl. Acad. Sci. 104(39), 15236–15241 (2007)CrossRefGoogle Scholar
  15. 15.
    Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theoret. Biol. 99, 237–247 (1982)CrossRefGoogle Scholar
  16. 16.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, Caltech (1998)Google Scholar
  17. 17.
    Winslow, A.: Staged self-assembly and polyomino context-free grammars. Nat. Comput. 14(2), 293–302 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Cameron Chalk
    • 1
  • Eric Martinez
    • 2
  • Robert Schweller
    • 2
  • Luis Vega
    • 2
  • Andrew Winslow
    • 2
  • Tim Wylie
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of Texas at AustinAustinUSA
  2. 2.Department of Computer ScienceUniversity of Texas - Rio Grande ValleyEdinburgUSA

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