When a Dollar Makes a BWT

TheBurrows-Wheeler-Transform(BWT)isareversiblestring transformation which plays a central role in text compression and is fun- damental in many modern bioinformatics applications. The BWT is a permutation of the characters, which is in general better compressible and allows to answer several different query types more efficiently than the original string. It is easy to see that not every string is a BWT image, and exact charac- terizations of BWT images are known. We investigate a related combi- natorial question. In many applications, a sentinel character $is added to mark the end of the string, and thus the BWT of a string ending with$ contains exactly one $character. We ask, given a string w, in which positions, if any, can the$-character be inserted to turn w into the BWT image of a word ending with the sentinel character. We show that this depends only on the standard permutation of w and give a combinatorial characterization of such positions via this permutation. We then develop an O(n log n)-time algorithm for identifying all such positions, improving on the naive quadratic time algorithm