Prefix Reversals on Binary and Ternary Strings

Given a permutation $\pi$, the application of prefix reversal $f^{(i)}$ to $\pi$ reverses the order of the first $i$ elements of $\pi$. The problem of sorting by prefix reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Discrete Math., 27 (1979), 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.