ar X iv : m at h / 06 02 45 6 v 1 [ m at h . C O ] 2 1 Fe b 20 06 Prefix reversals on binary and ternary strings ⋆

The problem Sorting By Prefix Reversals (also known as pancake flipping), made famous by Gates and Papadimitriou [7], 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 demonstrate polynomial-time algorithms for optimally sorting binary strings and ternary strings with prefix reversals; we demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; we give an exact expression for the prefix reversal diameter of binary strings, and we give bounds on the diameter of ternary strings. We also define an interesting weakening of sorting on strings, called grouping (where alike symbols have only to be brought together, with no restraint on symbol order) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed.