Prefix Reversals on Binary and Ternary Strings
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.
KeywordsBinary String Size Alphabet Distance Problem Sorting Problem Identical Symbol
Unable to display preview. Download preview PDF.
- 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
- Goldstein, A., Kolman, P., Zheng, J.: Minimum Common String Partition problem: hardness and approximations, Electron. J. Combin. 12, #R50 (2005)Google Scholar
- 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
- 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
- Scott, A.D.: Personal communicationGoogle Scholar