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Molecular computing, bounded nondeterminism, and efficient recursion

  • Richard Beigel
  • Bin Fu
Session 21: Biocomputing
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)

Abstract

The maximum number of strands used is an important measure of a molecular algorithm's complexity. This measure is also called the space used by the algorithm. We show that every NP problem that can be solved with b(n) bits of nondeterminism can be solved by molecular computation in a polynomial number of steps, with four test tubes, in space 2b(n. In addition, we identify a large class of recursive algorithms that can be implemented using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3-SAT, independent set, and 3-colorability.

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References

  1. 1.
    L. Adleman. Molecular computation of solutions to combinatorial problems. Science, 266:1021–1024, Nov. 1994.PubMedGoogle Scholar
  2. 2.
    L. Adleman. On constructing a molecular computer. In 1st DIMACS workshop on DNA Computing, 1995.Google Scholar
  3. 3.
    E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proc. 11th Ann. Conf. Structure in Complexity Theory, pp. 290–299, 1996.Google Scholar
  4. 4.
    D. Beaver. A universal molecular computer. CSE 95-001, Penn. State Univ., 1995.Google Scholar
  5. 5.
    R. Beigel. Maximum independent set algorithms. Manuscript, 1996.Google Scholar
  6. 6.
    R. Beigel and D. Eppstein. 3-coloring in time O(1.3446n): a no-MIS algorithm. In Proc. 36th IEEE FOCS, pp. 444–452, 1995.Google Scholar
  7. 7.
    R. Beigel and J. Goldsmith. Downward separation fails catastrophically for limited nondeterminism classes. In Proc. 9th Ann. Conf. Structure in Complexity Theory, pp. 134–138, 1994.Google Scholar
  8. 8.
    D. Boneh, C. Dunworth, R. J. Lipton, and J. Sgall. On the computational power of DNA. Manuscript, 1996.Google Scholar
  9. 9.
    J. F. Buss and J. Goldsmith. Nondeterminism within P. SICOMP, 22:560–572, 1993.Google Scholar
  10. 10.
    J. D. C. Àlvarez and J. Torán. Complexity classes with complete problems between P and NP-complete. In Foundations of Computation Theory, pp. 13–24. Springer-Verlag, 1989. LNCS 380.Google Scholar
  11. 11.
    J. Díaz and J. Torán. Classes of bounded nondeterminism. MST, 23:21–32, 1990.Google Scholar
  12. 12.
    J. Goldsmith, M. Levy, and M. Mundhenk. Limited nondeterminism. SIGACT News, pp. 20–29, June 1996.Google Scholar
  13. 13.
    L. Hemachandra and S. Jha. Defying upward and downward separation. In Proc. 10th STACS, pp. 185–195. Springer-Verlag, 1993. LNCS 665.Google Scholar
  14. 14.
    C. M. R. Kintala. Computations with a restricted number of nondeterministic steps. PhD thesis, Penn. State Univ., University Park, PA, 1977.Google Scholar
  15. 15.
    C. M. R. Kintala and P. C. Fischer. Computations with a restricted number of nondeterministic steps. In Proc. 9th A CM STOC, pp. 178–185, 1977.Google Scholar
  16. 16.
    C. M. R. Kintala and P. C. Fischer. Refining nondeterminism in relativized polynomial-time bounded computations. SICOMP, 9(1):46–53, Feb. 1980.Google Scholar
  17. 17.
    R. Lipton. Using DNA to solve NP-complete problems. Science, 268:542–545, Apr. 1995.PubMedGoogle Scholar
  18. 18.
    B. Monien and E. Speckenmeyer. Solving satisfiability in less than 2n steps. Discrete Appl. Math., 10:287–295, 1985.CrossRefGoogle Scholar
  19. 19.
    M. Ogihara. Breadth first search 3SAT algorithms for DNA computers. TR 629, U. Rochester, July 1996.Google Scholar
  20. 20.
    C. H. Papadimitriou and M. Yannakakis. On limited nondeterminism and the complexity of the V-C dimension. In Proc. 8th Ann. Conf. Structure in Complexity Theory, pp. 12–18, 1993.Google Scholar
  21. 21.
    N. Pippenger and M. Fischer. Relations among complexity measures. J. ACM, 26, 1979.Google Scholar
  22. 22.
    J. Robson. Algorithms for maximum independent sets. J. Algorithms, 7:425–440, 1986.CrossRefGoogle Scholar
  23. 23.
    D. Roos and K. Wagner. On the power of bio-computers. TR, U. of Wurzburg, Feb. 1995. ftp://haegar.informatik.uni-wuerzburg.de/pub/TRs/ro-wa95.ps.gz.Google Scholar
  24. 24.
    P. Rothemund. A DNA and restriction enzyme implementation of Turing machines. http://www.ugcs.caltech.edu/tt∼pwkr/oett.html.Google Scholar
  25. 25.
    L. Sanchis. Constructing language instances based on partial information. International Jour. Found. Comp. Sci., 5(2):209–229, 1994.CrossRefGoogle Scholar
  26. 26.
    I. Schiermeyer. Pure literal lookahead: An O(l,497n) 3-satisfiability algorithm. Manuscript, August 14, 1996.Google Scholar
  27. 27.
    C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proc. 3rd ICALP, pp. 322–337, 1976.Google Scholar
  28. 28.
    A. L. Selman. Natural self-reducible sets. TR, Northeastern Univ., 1986.Google Scholar
  29. 29.
    W. Smith and A. Schweitzer. DNA computers in vitro and vivo. TR, NEC, 1995.Google Scholar
  30. 30.
    R. Tarjan. Finding a maximum clique. TR 72-123, Cornell Univ., 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  1. 1.Yale University, University of Maryland, and Lehigh UniversityUSA
  2. 2.Yale University and University of MarylandUSA

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