Abstract
We consider a rearrangement problem of two-dimensional bicolor arrays by prefix reversals as a generalization of the burnt pancake problem. An equivalence relation on the set of bicolor arrays is induced by prefix reversals, and the rearrangement problem is to characterize the equivalence classes. While previously studied the rearrangement problem for unicolor arrays made use of the classical group theoretic tools, the present problem is quite different. For bicolor arrays a rearrangement can be described by partial injections, and thus we characterize the equivalence classes in terms of a groupoid action. We also outline an algorithm for rearrangement by prefix reversals and estimate a minimum number of rearrangements needed to rearrange bicolor arrays by prefix reversals.
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
References
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). https://doi.org/10.1007/3-540-48523-6_17
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). https://doi.org/10.1007/3-540-45749-6_21
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). https://doi.org/10.1007/978-3-642-32589-2_24
Chitturi, B., et al.: An \((18/11)n\) upper bound for rearranging by prefix reversals. Theoret. Comput. Sci. 410(36), 3372–3390 (2009)
Cohen, D.S., Blum, M.: On the problem of rearranging burnt pancakes. Discrete Appl. Math. 61(2), 105–120 (1995)
Gates, W., Papadimitriou, C.: Bounds for sorting by prefix reversal. Discrete Math. 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)
Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)
Rotman, J.J.: An Introduction to the Theory of Groups, 4th edn. Springer, New York (1994). https://doi.org/10.1007/978-1-4612-4176-8
Yamamura, A.: Rearranging two dimensional arrays by prefix reversals. In: Bojańczyk, M., Lasota, S., Potapov, I. (eds.) RP 2015. LNCS, vol. 9328, pp. 153–165. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24537-9_14
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Yamamura, A., Kase, R., Jajcayová, T.B. (2020). Groupoid Action and Rearrangement Problem of Bicolor Arrays by Prefix Reversals. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_31
Download citation
DOI: https://doi.org/10.1007/978-3-030-50026-9_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-50025-2
Online ISBN: 978-3-030-50026-9
eBook Packages: Computer ScienceComputer Science (R0)