Reversal Distances for Strings with Few Blocks or Small Alphabets

Abstract. We study the String Reversal Distance problem, an ex-tension of the well-known Sorting by Reversals problem. String Re-versal Distance takes two strings S and T as input, and asks for a min-imum number of reversals to obtain T from S. We consider four variants: String Reversal Distance, String Prefix Reversal Distance (in which any reversal must include the first letter of the string), and the signed variants of these problems, namely Signed String Reversal Distance and Signed String Prefix Reversal Distance. We study algorithmic properties of these four problems, in connection with two parameters of the input strings: the number of blocks they contain (a block being maximal substring such that all letters in the substring are equal), and the alphabet size Σ. For instance, we show that Signed String Reversal Distance and Signed String Prefix Reversal Distance are NP-hard even if the input strings have only one letter.