Pancake Flipping is the problem of sorting a stack of pancakes of different sizes (that is, a permutation), when the only allowed operation is to insert a spatula anywhere in the stack and to flip the pancakes above it (that is, to perform a prefix reversal). In the burnt variant, one side of each pancake is marked as burnt, and it is required to finish with all pancakes having the burnt side down. Computing the optimal scenario for any stack of pancakes and determining the worst-case stack for any stack size have been challenges over more than three decades. Beyond being an intriguing combinatorial problem in itself, it also yields applications, e.g. in parallel computing and computational biology.

In this paper, we show that the Pancake Flipping problem, in its original (unburnt) variant, is NP-hard, thus answering the long-standing question of its computational complexity.


Pancake problem Computational complexity Permutations Prefix reversals 


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  1. 1.
    Bafna, V., Pevzner, P.: Genome rearrangements and sorting by reversals. In: FOCS, pp. 148–157. IEEE (1993)Google Scholar
  2. 2.
    Berman, P., Hannenhalli, S., Karpinski, M.: 1.375-Approximation Algorithm for Sorting by Reversals. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 200–210. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Berman, P., Karpinski, M.: On Some Tighter Inapproximability Results (Extended Abstract). In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Chitturi, B., Fahle, W., Meng, Z., Morales, L., Shields, C.O., Sudborough, I., Voit, W.: An (18/11)n upper bound for sorting by prefix reversals. Theoretical Computer Science 410(36), 3372–3390 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cibulka, J.: On average and highest number of flips in pancake sorting. Theoretical Computer Science 412(8-10), 822–834 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cohen, D., Blum, M.: On the problem of sorting burnt pancakes. Discrete Applied Mathematics 61(2), 105–120 (1995)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dweighter, H. American Mathematics Monthly, 82(1) (1975): (pseudonym of Goodman, J.E.)Google Scholar
  8. 8.
    Fischer, J., Ginzinger, S.: 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. 9.
    Gates, W., Papadimitriou, C.: Bounds for sorting by prefix reversal. Discrete Mathematics 27(1), 47–57 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hannenhalli, S., Pevzner, P.: Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. In: STOC, pp. 178–189. ACM (1995)Google Scholar
  11. 11.
    Heydari, M., Sudborough, I.: On Sorting by Prefix Reversals and the Diameter of Pancake Networks. In: Meyer auf der Heide, F., Rosenberg, A.L., Monien, B. (eds.) Heinz Nixdorf Symposium 1992. LNCS, vol. 678, pp. 218–227. Springer, Heidelberg (1993)Google Scholar
  12. 12.
    Heydari, M., Sudborough, I.: On the diameter of the pancake network. Journal of Algorithms 25(1), 67–94 (1997)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Labarre, A., Cibulka, J.: Polynomial-time sortable stacks of burnt pancakes. Theoretical Computer Science 412(8-10), 695–702 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Laurent Bulteau
    • 1
  • Guillaume Fertin
    • 1
  • Irena Rusu
    • 1
  1. 1.Laboratoire d’Informatique de Nantes-Atlantique (LINA)UMR CNRS 6241, Université de NantesNantes Cedex 3France

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