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

Authors David Furcy, Scott M. Summers, Logan Withers



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Author Details

David Furcy
  • Computer Science Department, University of Wisconsin Oshkosh, WI, USA
Scott M. Summers
  • Computer Science Department, University of Wisconsin Oshkosh, WI, USA
Logan Withers
  • Computer Science Department, University of Wisconsin Oshkosh, WI, USA

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David Furcy, Scott M. Summers, and Logan Withers. Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D. In 27th International Conference on DNA Computing and Molecular Programming (DNA 27). Leibniz International Proceedings in Informatics (LIPIcs), Volume 205, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.DNA.27.4

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 k × 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 k-2 digits, represented in a geometrically staggered fashion within k rows. This nearly optimal digit density, along with the base of the counter, which is proportional to N^{1/(k-1)}, results in an upper bound of O(N^{1/(k-1)} + log N), 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, Ω(N^{1/k}), is also an asymptotic improvement over the previous state-of-the-art lower bound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Self-assembly
  • algorithmic self-assembly
  • tile self-assembly

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References

  1. Leonard M. Adleman, Qi Cheng, Ashish Goel, and Ming-Deh A. Huang. Running time and program size for self-assembled squares. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (STOC), pages 740-748, 2001. Google Scholar
  2. Gagan Aggarwal, Qi Cheng, Michael H. Goldwasser, Ming-Yang Kao, Pablo Moisset de Espanés, and Robert T. Schweller. Complexities for generalized models of self-assembly. SIAM Journal on Computing (SICOMP), 34:1493-1515, 2005. Google Scholar
  3. Matthew Cook, Yunhui Fu, and Robert T. Schweller. 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), pages 570-589, 2011. Google Scholar
  4. Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Mashhood Ishaque, Eynat Rafalin, Robert T. Schweller, and Diane L. Souvaine. Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues. Natural Computing, 7(3):347-370, 2008. URL: https://doi.org/10.1007/s11047-008-9073-0.
  5. David Doty, Matthew J. Patitz, Dustin Reishus, Robert T. Schweller, and Scott M. Summers. Strong fault-tolerance for self-assembly with fuzzy temperature. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pages 417-426, 2010. Google Scholar
  6. David Doty, Matthew J. Patitz, and Scott M. Summers. Limitations of self-assembly at temperature 1. Theoretical Computer Science, 412:145-158, 2011. Google Scholar
  7. Sándor P. Fekete, Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, and Robert T. Schweller. 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, pages 148-167, 2015. Google Scholar
  8. Bin Fu, Matthew J. Patitz, Robert T. Schweller, and Robert Sheline. Self-assembly with geometric tiles. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, pages 714-725, 2012. Google Scholar
  9. David Furcy, Samuel Micka, and Scott M. Summers. Optimal program-size complexity for self-assembled squares at temperature 1 in 3D. Algorithmica, 77(4):1240-1282, 2017. Google Scholar
  10. David Furcy and Scott M. Summers. Optimal self-assembly of finite shapes at temperature 1 in 3D. Algorithmica, 80(6):1909-1963, 2018. Google Scholar
  11. David Furcy, Scott M. Summers, and Christian Wendlandt. Self-assembly of and optimal encoding within thin rectangles at temperature-1 in 3D. Theoretical Computer Science, 872:55-78, 2021. Google Scholar
  12. Oscar Gilbert, Jacob Hendricks, Matthew J. Patitz, and Trent A. Rogers. 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, pages 937-956, 2016. Google Scholar
  13. Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, and Scott M. Summers. The power of duples (in self-assembly): It’s not so hip to be square. Theoretical Computer Science, 743:148-166, 2018. Google Scholar
  14. James I. Lathrop, Jack H. Lutz, and Scott M. Summers. Strict self-assembly of discrete Sierpinski triangles. Theoretical Computer Science, 410:384-405, 2009. Google Scholar
  15. Jack H. Lutz and Brad Shutters. Approximate self-assembly of the sierpinski triangle. Theory Comput. Syst., 51(3):372-400, 2012. Google Scholar
  16. Ján Manuch, Ladislav Stacho, and Christine Stoll. Two lower bounds for self-assemblies at temperature 1. Journal of Computational Biology, 17(6):841-852, 2010. Google Scholar
  17. P.-E. Meunier, M. J. Patitz, S. M. Summers, G. Theyssier, A. Winslow, and D. Woods. Intrinsic universality in tile self-assembly requires cooperation. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 752-771, 2014. Google Scholar
  18. Pierre-Étienne Meunier, Damien Regnault, and Damien Woods. 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, pages 727-737, 2020. Google Scholar
  19. Pierre-Étienne Meunier and Damien Woods. 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, pages 328-341, 2017. Google Scholar
  20. Matthew J. Patitz, Robert T. Schweller, and Scott M. Summers. 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, pages 175-189, Berlin, Heidelberg, 2011. Springer-Verlag. URL: http://dl.acm.org/citation.cfm?id=2042033.2042050.
  21. Paul W. K. Rothemund. Theory and Experiments in Algorithmic Self-Assembly. PhD thesis, University of Southern California, December 2001. Google Scholar
  22. Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares (extended abstract). In The Thirty-Second Annual ACM Symposium on Theory of Computing (STOC), pages 459-468, 2000. Google Scholar
  23. Nadrian C. Seeman. Nucleic-acid junctions and lattices. Journal of Theoretical Biology, 99:237-247, 1982. Google Scholar
  24. David Soloveichik and Erik Winfree. Complexity of self-assembled shapes. SIAM Journal on Computing (SICOMP), 36(6):1544-1569, 2007. Google Scholar
  25. Hao Wang. Proving theorems by pattern recognition - II. The Bell System Technical Journal, XL(1):1-41, 1961. Google Scholar
  26. Erik Winfree. Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, June 1998. Google Scholar
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