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Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D

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Abstract

We investigate a fundamental question regarding a benchmark class of shapes in one of the simplest, yet most widely utilized abstract models of algorithmic tile self-assembly. More specifically, we study the directed tile complexity of a \(\varvec{k}\times \varvec{N}\) thin rectangle in Winfree’s ubiquitous abstract Tile Assembly Model, assuming that cooperative binding cannot be enforced (temperature-1 self-assembly) and that tiles are allowed to be placed at most one step into the third dimension (just-barely 3D). While the directed tile complexities of a square and a scaled-up version of any algorithmically specified shape at temperature 1 in just-barely 3D are both asymptotically the same as they are (respectively) at temperature 2 in 2D, the (nearly tight) bounds on the directed tile complexity of a thin rectangle at temperature 2 in 2D are not currently known to hold at temperature 1 in just-barely 3D. Motivated by this discrepancy, we establish new lower and upper bounds on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D. The proof of our upper bound is based on the construction of a novel, just-barely 3D temperature-1 self-assembling counter. Each value of the counter is comprised of \(\varvec{k} - {\textbf {2}}\) digits, represented in a geometrically staggered fashion within \(\varvec{k}\) rows. This nearly optimal digit density, along with the base of the counter, which is proportional to \(\varvec{N}^{\frac{\varvec{1}}{\varvec{k}-{\textbf {1}}}}\), results in an upper bound on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D of \(\varvec{O}\left( \varvec{N}^{\frac{{\textbf {1}}}{\varvec{k}-{\textbf {1}}}} + \varvec{k} \right) \), and is an asymptotic improvement over the previous state-of-the-art upper bound. On our way to proving our lower bound, we develop a new, more powerful type of specialized Window Movie Lemma that lets us bound the number of “sufficiently similar” ways to assign glues to a set (rather than a sequence) of fixed locations. Consequently, our lower bound on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D of \(\varvec{\Omega }\left( \varvec{N}^{\frac{{\textbf {1}}}{\varvec{k}}} \right) \), is also an asymptotic improvement over the previous state-of-the-art lower bound.

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References

  1. Winfree, E.: Algorithmic self-assembly of DNA. PhD thesis, California Institute of Technology (1998)

  2. Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

  3. Wang, H.: Proving theorems by pattern recognition – II. Bell Syst. Tech. J. XL(1), 1–41 (1961)

  4. Rothemund, P.W.K., Winfree, E.: The program-size complexity of selfassembled squares (extended abstract). In: The Thirty-Second Annual ACM Symposium on Theory of Computing (STOC), pp. 459–468 (2000)

  5. Manuch, J., Stacho, L., Stoll, C.: Two lower bounds for self-assemblies at temperature 1. J. Comput. Biol. 17(6), 841–852 (2010)

    Article  MathSciNet  Google Scholar 

  6. Meunier, P.-E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 752–771 (2014)

  7. Meunier, P., Woods, D.: The non-cooperative tile assembly model is not intrinsically universal or capable of bounded turing machine simulation. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19–23, 2017, pp. 328–341 (2017)

  8. Meunier, P., Regnault, D., Woods, D.: The program-size complexity of self-assembled paths. In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pp. 727–737 (2020)

  9. Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theor. Comput. Sci. 412, 145–158 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doty, D., Patitz, M.J., Reishus, D., Schweller, R.T., Summers, S.M.: Strong fault-tolerance for self-assembly with fuzzy temperature. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 417–426 (2010)

  12. Fekete, S.P., Hendricks, J., Patitz, M.J., Rogers, T.A., Schweller, R.T.: Universal computation with arbitrary polyomino tiles in non-cooperative self-assembly. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pp. 148–167 (2015)

  13. Fu, B., Patitz, M.J., Schweller, R.T., Sheline, R.: Self-assembly with geometric tiles. In: Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, pp. 714–725 (2012)

  14. Gilbert, O., Hendricks, J., Patitz, M.J., Rogers, T.A.: Computing in continuous space with self-assembling polygonal tiles (extended abstract). In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pp. 937–956 (2016)

  15. Patitz, M.J., Schweller, R.T., Summers, S.M.: Exact shapes and Turing universality at temperature 1 with a single negative glue. In: Proceedings of the 17th International Conference on DNA Computing and Molecular Programming. DNA’11, pp. 175–189. Springer, Berlin, Heidelberg (2011). http://dl.acm.org/citation.cfm?id=2042033.2042050

  16. Hendricks, J., Patitz, M.J., Rogers, T.A., Summers, S.M.: The power of duples (in self-assembly): It’s not so hip to be square. Theor. Comput. Sci. 743, 148–166 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cook, M., Fu, Y., Schweller, R.T.: Temperature 1 self-assembly: Deterministic assembly in 3D and probabilistic assembly in 2D. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 570–589 (2011)

  18. Furcy, D., Micka, S., Summers, S.M.: Optimal program-size complexity for self-assembled squares at temperature 1 in 3D. Algorithmica 77(4), 1240–1282 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Furcy, D., Summers, S.M., Wendlandt, C.: Self-assembly of and optimal encoding within thin rectangles at temperature-1 in 3D. Theor. Comput. Sci. 872, 55–78 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Furcy, D., Summers, S.M.: Optimal self-assembly of finite shapes at temperature 1 in 3D. Algorithmica 80(6), 1909–1963 (2018)

  21. Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.A.: Running time and program size for self–assembled squares. In: Proceedings of the Thirty–Third Annual ACM Symposium on Theory of Computing (STOC), pp. 740–748 (2001)

  22. Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. (SICOMP) 36(6), 1544–1569 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Aggarwal, G., Cheng, Q., Goldwasser, M.H., Kao, M.-Y., de Espanés, P.M., Schweller, R.T.: Complexities for generalized models of selfassembly. SIAM J. Comput. (SICOMP) 34, 1493–1515 (2005)

    Article  MATH  Google Scholar 

  24. Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theor. Comput. Sci. 410, 384–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rothemund, P.W.K.: Theory and experiments in algorithmic selfassembly. PhD thesis, University of Southern California (2001)

  26. Lutz, J.H., Shutters, B.: Approximate self-assembly of the sierpinski triangle. Theor. Comput. Syst. 51(3), 372–400 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Computer Science Department, University of Wisconsin Oshkosh, 800 Algoma Blvd, Oshkosh, 54901, WI, USA

    David Furcy, Scott M. Summers & Logan Withers

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  1. David Furcy
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  2. Scott M. Summers
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Correspondence to Scott M. Summers.

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David Furcy, Scott M. Summers and Logan Withers contributed equally to this work.

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Furcy, D., Summers, S.M. & Withers, L. Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D. Theory Comput Syst 67, 1082–1130 (2023). https://doi.org/10.1007/s00224-023-10137-9

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