Reducing the Worst Case Running Times of a Family of RNA and CFG Problems, Using Valiant’s Approach

  • Shay Zakov
  • Dekel Tsur
  • Michal Ziv-Ukelson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6293)


We study Valiant’s classical algorithm for Context Free Grammar recognition in sub-cubic time, and extract features that are common to problems on which Valiant’s approach can be applied. Based on this, we describe several problem templates, and formulate generic algorithms that use Valiant’s technique and can be applied to all problems which abide by these templates. These algorithms obtain new worst case running time bounds for a large family of important problems within the world of RNA Secondary Structures and Context Free Grammars.


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  1. 1.
    Consortium, A.F.B., Backofen, R., Bernhart, S.H., Flamm, C., Fried, C., Fritzsch, G., Hackermuller, J., Hertel, J., Hofacker, I.L., Missal, K., Mosig, A., Prohaska, S.J., Rose, D., Stadler, P.F., Tanzer, A., Washietl, S., Will, S.: RNAs everywhere: genome-wide annotation of structured RNAs. J. Exp. Zoolog. B. Mol. Dev. Evol. 308, 1–25 (2007)Google Scholar
  2. 2.
    Nussinov, R., Jacobson, A.B.: Fast algorithm for predicting the secondary structure of single-stranded RNA. PNAS 77, 6309–6313 (1980)CrossRefPubMedPubMedCentralGoogle Scholar
  3. 3.
    Zuker, M., Stiegler, P.: Optimal computer folding of large RNA sequences using thermodynamics and auxiliary information. Nucleic Acids Research 9, 133–148 (1981)CrossRefPubMedPubMedCentralGoogle Scholar
  4. 4.
    Alkan, C., Karakoç, E., Nadeau, J.H., Sahinalp, S.C., Zhang, K.: RNA-RNA interaction prediction and antisense RNA target search. Journal of Computational Biology 13, 267–282 (2006)CrossRefPubMedGoogle Scholar
  5. 5.
    McCaskill, J.S.: The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers 29, 1105–1119 (1990)CrossRefPubMedGoogle Scholar
  6. 6.
    Bernhart, S., Tafer, H., Mückstein, U., Flamm, C., Stadler, P., Hofacker, I.: Partition function and base pairing probabilities of RNA heterodimers. Algorithms for Molecular Biology 1, 3 (2006)CrossRefPubMedPubMedCentralGoogle Scholar
  7. 7.
    Chitsaz, H., Salari, R., Sahinalp, S.C., Backofen, R.: A partition function algorithm for interacting nucleic acid strands. Bioinformatics 25, i365–i373 (2009)Google Scholar
  8. 8.
    Zhang, K.: Computing similarity between RNA secondary structures. In: INTSYS 1998: Proceedings of the IEEE International Joint Symposia on Intelligence and Systems, p. 126. IEEE Computer Society, Washington (1998)Google Scholar
  9. 9.
    Sankoff, D.: Simultaneous solution of the RNA folding, alignment and protosequence problems. SIAM Journal on Applied Mathematics 45, 810–825 (1985)CrossRefGoogle Scholar
  10. 10.
    Sakakibara, Y., Brown, M., Hughey, R., Mian, I., Sjolander, K., Underwood, R., Haussler, D.: Stochastic context-free grammers for tRNA modeling. Nucleic Acids Research 22, 5112 (1994)CrossRefPubMedPubMedCentralGoogle Scholar
  11. 11.
    Teitelbaum, R.: Context-free error analysis by evaluation of algebraic power series. In: STOC, pp. 196–199. ACM, New York (1973)Google Scholar
  12. 12.
    Dowell, R., Eddy, S.: Evaluation of several lightweight stochastic context-free grammars for RNA secondary structure prediction. BMC bioinformatics 5, 71 (2004)CrossRefPubMedPubMedCentralGoogle Scholar
  13. 13.
    Do, C.B., Woods, D.A., Batzoglou, S.: CONTRAfold: RNA secondary structure prediction without physics-based models. Bioinformatics 22, e90–e98 (2006)Google Scholar
  14. 14.
    Cocke, J., Schwartz, J.T.: Programming Languages and Their Compilers. Courant Institute of Mathematical Sciences, New York (1970)Google Scholar
  15. 15.
    Kasami, T.: An efficient recognition and syntax analysis algorithm for context-free languages. Technical Report AFCRL-65-758, Air Force Cambridge Res. Lab., Bedford Mass. (1965)Google Scholar
  16. 16.
    Younger, D.H.: Recognition and parsing of context-free languages in time n 3. Information and Control 10, 189–208 (1967)CrossRefGoogle Scholar
  17. 17.
    Valiant, L.: General context-free recognition in less than cubic time. Journal of Computer and System Sciences 10, 308–315 (1975)CrossRefGoogle Scholar
  18. 18.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)CrossRefGoogle Scholar
  19. 19.
    Akutsu, T.: Approximation and exact algorithms for RNA secondary structure prediction and recognition of stochastic context-free languages. Journal of Combinatorial Optimization 3, 321–336 (1999)CrossRefGoogle Scholar
  20. 20.
    Benedí, J., Sánchez, J.: Fast Stochastic Context-Free Parsing: A Stochastic Version of the Valiant Algorithm. In: Martí, J., Benedí, J.M., Mendonça, A.M., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4477, pp. 80–88. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  21. 21.
    Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. In: STOC 2007: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pp. 590–598. ACM, New York (2007)CrossRefGoogle Scholar
  22. 22.
    Graham, S.L., Harrison, M.A., Ruzzo, W.L.: An improved context-free recognizer. ACM Transactions on Programming Languages and Systems 2, 415–462 (1980)CrossRefGoogle Scholar
  23. 23.
    Arlazarov, V.L., Dinic, E.A., Kronod, M.A., Faradzev, I.A.: On economical construction of the transitive closure of an oriented graph. Soviet. Math. Dokl. 11, 1209–1210 (1970)Google Scholar
  24. 24.
    Frid, Y., Gusfield, D.: A simple, practical and complete O\((\frac{n^3}{ \log n})\)-time algorithm for RNA folding using the four-russians speedup. In: Salzberg, S.L., Warnow, T. (eds.) WABI 2009. LNCS, vol. 5724, pp. 97–107. Springer, Heidelberg (2009)Google Scholar
  25. 25.
    Klein, D., Manning, C.D.: A* parsing: Fast exact viterbi parse selection. In: HLT-NAACL, pp. 119–126 (2003)Google Scholar
  26. 26.
    Jansson, J., Ng, S., Sung, W., Willy, H.: A faster and more space-efficient algorithm for inferring arc-annotations of RNA sequences through alignment. Algorithmica 46, 223–245 (2006)CrossRefGoogle Scholar
  27. 27.
    Wexler, Y., Zilberstein, C.B.Z., Ziv-Ukelson, M.: A study of accessible motifs and RNA folding complexity. Journal of Computational Biology 14, 856–872 (2007)CrossRefPubMedGoogle Scholar
  28. 28.
    Ziv-Ukelson, M., Gat-Viks, I., Wexler, Y., Shamir, R.: A faster algorithm for RNA co-folding. In: Crandall, K.A., Lagergren, J. (eds.) WABI 2008. LNCS (LNBI), vol. 5251, pp. 174–185. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    Backofen, R., Tsur, D., Zakov, S., Ziv-Ukelson, M.: Sparse RNA folding: Time and space efficient algorithms. In: Kucherov, G., Ukkonen, E. (eds.) CPM 2009. LNCS, vol. 5577, pp. 249–262. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  30. 30.
    Salari, R., Mohl, M., Will, S., Sahinalp, S., Backofen, R.: Time and Space Efficient RNA-RNA Interaction Prediction via Sparse Folding. In: Berger, B. (ed.) RECOMB 2010. LNCS, vol. 6044, pp. 473–490. Springer, Heidelberg (2010)Google Scholar
  31. 31.
    Havgaard, J., Lyngso, R., Stormo, G., Gorodkin, J.: Pairwise local structural alignment of RNA sequences with sequence similarity less than 40%. Bioinformatics 21, 1815–1824 (2005)CrossRefPubMedGoogle Scholar
  32. 32.
    Will, S., Reiche, K., Hofacker, I.L., Stadler, P.F., Backofen, R.: Inferring non-coding RNA families and classes by means of genome-scale structure-based clustering. PLOS Computational Biology 3, 65 (2007)CrossRefGoogle Scholar
  33. 33.
    Baker, J.K.: Trainable grammars for speech recognition. The Journal of the Acoustical Society of America 65, S132 (1979)CrossRefGoogle Scholar
  34. 34.
    Goto, K., Geijn, R.: Anatomy of high-performance matrix multiplication. ACM Transactions on Mathematical Software (TOMS) 34, 1–25 (2008)CrossRefGoogle Scholar
  35. 35.
    Robinson, S.: Toward an optimal algorithm for matrix multiplication. News Journal of the Society for Industrial and Applied Mathematics 38 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Shay Zakov
    • 1
  • Dekel Tsur
    • 1
  • Michal Ziv-Ukelson
    • 1
  1. 1.Department of Computer ScienceBen-Gurion University of the NegevIsrael

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