Prefix Reversals on Binary and Ternary Strings

  • Cor Hurkens
  • Leo van Iersel
  • Judith Keijsper
  • Steven Kelk
  • Leen Stougie
  • John Tromp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4545)

Abstract

Given a permutation π, the application of prefix reversal f(i) to π reverses the order of the first i elements of π. The problem of Sorting By Prefix Reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Bounds for sorting by prefix reversal, Discrete Mathematics 27, pp. 47-57), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings, and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bader, M., Ohlebusch, E.: Sorting by weighted reversals, transpositions, and inverted transpositions. In: Apostolico, A., Guerra, C., Istrail, S., Pevzner, P., Waterman, M. (eds.) RECOMB 2006. LNCS (LNBI), vol. 3909, pp. 563–577. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. Bergeron, A., Mixtacki, J., Stoye, J.: On sorting by translocations. J. Comp. Biol. 13(2), 567–578 (2006)CrossRefMathSciNetGoogle Scholar
  3. Chen, X., Zheng, J., Fu, Z., Nan, P., Zhong, Y., Lonardi, S., Jiang, T.: Assignment of orthologous genes via genome rearrangement, IEEE/ACM Trans. Comput. Biol. Bioinform. 2(4) (October- December 2005)Google Scholar
  4. Christie, D.A., Irving, R.W.: Sorting strings by reversals and transpositions. SIAM J. on Discrete Math. 14(2), 193–206 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. Cohen, D.S., Blum, M.: On the problem of sorting burnt pancakes. Discrete Appl. Math. 61(2), 105–120 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. Elias, I., Hartman, T.: A 1.375-approximation algorithm for sorting by transpositions. IEEE/ACM Trans. Comput. Biology Bioinform. 3(4), 369–379 (2006)CrossRefGoogle Scholar
  7. Eriksson, H., Eriksson, K., Karlander, J., Svensson, L., Wastlund, J.: Sorting a bridge hand. Discrete Math. 241, 289–300 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. Fischer, J., Ginzinger, S.W.: A 2-approximation algorithm for sorting by prefix reversals. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 415–425. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. Garey, M.R., Johnson, D.S.: Complexity results for multiprocessor scheduling under resource constraints. SIAM J. Comput. 4(4), 397–411 (1975)MATHCrossRefMathSciNetGoogle Scholar
  10. Garey, M.R., Johnson, D.S.: Computers And Intractability: A Guide To The Theory Of NP-completeness. W.H. Freeman, San Francisco, CA (1979)MATHGoogle Scholar
  11. Gates, W.H., Papadimitriou, C.H.: Bounds for sorting by prefix reversal. Discrete Math. 27, 47–57 (1979)CrossRefMathSciNetGoogle Scholar
  12. Goldstein, A., Kolman, P., Zheng, J.: Minimum Common String Partition problem: hardness and approximations, Electron. J. Combin. 12, #R50 (2005)Google Scholar
  13. Hannenhalli, S., Pevzner, P.A.: Transforming cabbage into turnip: a polynomial algorithm for sorting permutations by reversals. Jour. ACM 46(1), 1–27 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. Heydari, M.H., Sudborough, I.H.: On the diameter of the pancake network. J. Algorithms 25, 67–94 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. Hurkens, C.A.J., van Iersel, L.J.J., Keijsper, J.C.M., Kelk, S.M., Stougie, L., Tromp, J.T.: Prefix reversals on binary and ternary strings, technical report (2006), http://www.win.tue.nl/bs/spor/2006-10.pdf
  16. Morales, L., Sudborough, I.H.: Comparing Star and Pancake Networks. In: Mogensen, T.Æ., Schmidt, D.A., Sudborough, I.H. (eds.) The Essence of Computation. LNCS, vol. 2566, pp. 18–36. Springer, Heidelberg (2002)Google Scholar
  17. Radcliffe, A.J., Scott, A.D., Wilmer, E.L.: Reversals and transpositions over finite alphabets. SIAM J. on Discrete Math. 19(1), 224–244 (2005)MATHCrossRefMathSciNetGoogle Scholar
  18. Scott, A.D.: Personal communicationGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Cor Hurkens
    • 1
  • Leo van Iersel
    • 1
  • Judith Keijsper
    • 1
  • Steven Kelk
    • 2
  • Leen Stougie
    • 1
    • 2
  • John Tromp
    • 2
  1. 1.Technische Universiteit Eindhoven (TU/e), Den Dolech 2, 5612 AX EindhovenNetherlands
  2. 2.Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ AmsterdamNetherlands

Personalised recommendations