Skip to main content
SpringerLink
Log in

Constant-Size Tileset for Solving an NP-Complete Problem in Nondeterministic Linear Time

  • Conference paper
DNA Computing (DNA 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4848))

Included in the following conference series:

Abstract

The tile assembly model, a formal model of crystal growth, is of special interest to computer scientists and mathematicians because it is universal [1]. Therefore, tile assembly model systems can compute all the functions that computers compute. In this paper, I formally define what it means for a system to nondeterministically decide a set, and present a system that solves an NP-complete problem called SubsetSum. Because of the nature of NP-complete problems, this system can be used to solve all NP problems in polynomial time, with high probability. While the proof that the tile assembly model is universal [2] implies the construction of such systems, those systems are in some sense “large” and “slow.” The system presented here uses 49 = Θ(1) different tiles and computes in time linear in the input size. I also propose how such systems can be leveraged to program large distributed software systems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Winfree, E.: Simulations of computing by self-assembly of DNA. Technical Report CS-TR:1998:22, California Insitute of Technology, Pasadena, CA, USA (1998)

    Google Scholar 

  2. Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Insitute of Technology, Pasadena, CA, USA (June 1998)

    Google Scholar 

  3. Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266, 1021–1024 (1994)

    Article  Google Scholar 

  4. Braich, R., Johnson, C.R., Rothemund, P.W.K., Hwang, D., Chelyapov, N., Adleman, L.: Solution of a satisfiability problem on a gel-based DNA computer. In: Proceedings of DNA Computing: 6th International Workshop on DNA-Based Computers (DNA 2000), Leiden, The Netherlands, pp. 27–38 (June 2000)

    Google Scholar 

  5. Braich, R., Chelyapov, N., Johnson, C.R., Rothemund, P.W.K., Adleman, L.: Solution of a 20-variable 3-SAT problem on a DNA computer. Science 296(5567), 499–502 (2002)

    Article  Google Scholar 

  6. Winfree, E.: On the computational power of DNA annealing and ligation. DNA Based Computers 199–221 (1996)

    Google Scholar 

  7. Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), Madison, WI, USA, June 2003, vol. 2943, pp. 126–144. IEEE Computer Society Press, Los Alamitos (2003)

    Google Scholar 

  8. Baryshnikov, Y., Coffman, E.G., Seeman, N., Yimwadsana, T.: Self correcting self assembly: Growth models and the hammersley process. In: Carbone, A., Pierce, N.A. (eds.) DNA Computing. LNCS, vol. 3892, Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  9. Chen, H.L., Goel, A.: Error free self-assembly with error prone tiles. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA Computing. LNCS, vol. 3384, Springer, Heidelberg (2005)

    Google Scholar 

  10. Reif, J.H., Sahu, S., Yin, P.: Compact error-resilient computational DNA tiling assemblies. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA Computing. LNCS, vol. 3384, Springer, Heidelberg (2005)

    Google Scholar 

  11. Winfree, E.: Self-healing tile sets. Nanotechnology: Science and Computation, 55–78 (2006)

    Google Scholar 

  12. Barish, R., Rothemund, P.W.K., Winfree, E.: Two computational primitives for algorithmic self-assembly: Copying and counting. Nano Letters 5(12), 2586–2592 (2005)

    Article  Google Scholar 

  13. Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2(12), 424 (2004)

    Article  Google Scholar 

  14. Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Wasserman, H.: Linear self-assemblies: Equilibria, entropy, and convergence rates. In: Proceedings of the 6th International Conference on Difference Equations and Applications (ICDEA 2001), Augsburg, Germany (June 2001)

    Google Scholar 

  15. Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 2000, Portland, OR, USA, pp. 459–468. ACM Press, New York (2000)

    Chapter  Google Scholar 

  16. Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Kempe, D., de Espanes, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 2002), Montreal, Quebec, Canada, pp. 23–32. ACM Press, New York (2002)

    Chapter  Google Scholar 

  17. Adleman, L., Goel, A., Huang, M.-D., de Espanes, P.M.: Running time and program size for self-assembled squares. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 2002), Montreal, Quebec, Canada, pp. 740–748. ACM Press, New York (2001)

    Google Scholar 

  18. de Espanes, P.M.: Computerized exhaustive search for optimal self-assembly counters. In: Proceedings of the 2nd Foundations of Nanoscience: Self-Assembled Architectures and Devices (FNANO 2005), Snowbird, UT, USA, pp. 24–25 (April 2005)

    Google Scholar 

  19. Lagoudakis, M.G., LaBean, T.H.: 2D DNA self-assembly for satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 54, 141–154 (1999)

    MathSciNet  Google Scholar 

  20. Fu, T.J., Seeman, N.C.: DNA double-crossover molecules. Biochemistry 32(13), 3211–3220 (1993)

    Article  Google Scholar 

  21. Wang, H.: Proving theorems by pattern recognition. II. Bell System Technical Journal 40, 1–42 (1961)

    Google Scholar 

  22. Brun, Y.: Arithmetic computation in the tile assembly model: Addition and multiplication. Theoretical Computer Science 378, 17–31 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Brun, Y.: Nondeterministic polynomial time factoring in the tile assembly model. Theoretical Computer Science (2007), doi:10.1016/j.tcs.2007.07.051

    Google Scholar 

  24. Brun, Y.: Solving NP-complete problems in the tile assembly model. Theoretical Computer Science (2007), doi:10.1016/j.tcs.2007.07.052

    Google Scholar 

  25. Brun, Y., Medvidovic, N.: An architectural style for solving computationally intensive problems on large networks. In: Proceedings of Software Engineering for Adaptive and Self-Managing Systems (SEAMS 2007), Minneapolis, MN, USA (May 2007)

    Google Scholar 

  26. Brun, Y., Medvidovic, N.: Fault and adversary tolerance as an emergent property of distributed systems’ software architectures. In: Proceedings of the 2nd International Workshop on Engineering Fault Tolerant Systems (EFTS 2007), Dubrovnik, Croatia (September 2007)

    Google Scholar 

  27. Brun, Y.: Discreetly distributing computation via self-assembly. Technical Report USC-CSSE-2007-714, Center for Software Engineering, University of Southern California (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Southern California, Los Angeles, CA, 90089,  

    Yuriy Brun

Authors
  1. Yuriy Brun
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Max H. Garzon Hao Yan

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brun, Y. (2008). Constant-Size Tileset for Solving an NP-Complete Problem in Nondeterministic Linear Time. In: Garzon, M.H., Yan, H. (eds) DNA Computing. DNA 2007. Lecture Notes in Computer Science, vol 4848. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77962-9_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-77962-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-77961-2

  • Online ISBN: 978-3-540-77962-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation