Abstract
Generalizing the pancake sorting problem, we consider a reachability problem which asks whether an arbitrary two dimensional array can be obtained from an initial array by prefix reversals. In the case of the pancake sorting problem, sorting is always possible, whereas, it is not clear whether a rearrangement of two dimensional arrays is always possible. We shall prove any array is reachable from the initial array by prefix reversals unless the numbers of both rows and columns are divisible by 4. Using group theory, we also give a necessary and sufficient condition that an array is reachable from the initial array in such a case. We also give upper bounds on the number of prefix reversals to rearrange.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berman, P., Karpinski, M.: On some tighter inapproximability results. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999)
Berman, P., Hannenhalli, S., Karpinski, M.: 1.375-approximation algorithm for sorting by reversals. In: Möhring, R., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 200–210. Springer, Heidelberg (2002)
Bulteau, L., Fertin, G., Rusu, I.: Pancake flipping is hard. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 247–258. Springer, Heidelberg (2012)
Chitturi, B., Fahle, W., Meng, Z., Morales, L., Shields, C.O., Sudborough, I.H., Voit, W.: An \((18/11)n\) upper bound for rearranging by prefix reversals. Theoretical Computer Science 410(36), 3372–3390 (2009)
Cohen, D.S., Blum, M.: On the problem of rearranging burnt pancakes. Discrete Applied Mathematics 61(2), 105–120 (1995)
Gates, W., Papadimitriou, C.: Bounds for Sorting by Prefix Reversal. Discrete Mathematics 79, 47–57 (1979)
Heydari, M.H., Sudborough, I.H.: On the diameter of the pancake network. J. Algorithms 25(1), 67–94 (1997)
Kaplan, H., Shamir, R., Tarjan, R.E.: Faster and simpler algorithm for sorting signed permutations by reversals. In: ACM-SIAM SODA 1997, pp. 178–187 (1997)
Rotman, J.J.: An Introduction to the Theory of Groups, 4th edn. Springer (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Yamamura, A. (2015). Rearranging Two Dimensional Arrays by Prefix Reversals. In: Bojanczyk, M., Lasota, S., Potapov, I. (eds) Reachability Problems. RP 2015. Lecture Notes in Computer Science(), vol 9328. Springer, Cham. https://doi.org/10.1007/978-3-319-24537-9_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-24537-9_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24536-2
Online ISBN: 978-3-319-24537-9
eBook Packages: Computer ScienceComputer Science (R0)