Block Sorting-Based Transformations on Words: Beyond the Magic BWT

The Burrows-Wheeler Transform (BWT) is a word transformation introduced in 1994 for Data Compression and later results have contributed to make it a fundamental tool for the design of self-indexing compressed data structures. The Alternating Burrows-Wheeler Transform (ABWT) is a more recent transformation, studied in the context of Combinatorics on Words, that works in a similar way, using an alternating lexicographical order instead of the usual one. In this paper we study a more general class of block sorting-based transformations. The transformations in this new class prove to be interesting combinatorial tools that offer new research perspectives. In particular, we show that all the transformations in this class can be used as booster for memoryless compressors and we provide an upper bound on the number of equal-letter runs in their output. Moreover, we introduce the notion of rank-invertibility, a property related to the implementation of an efficient inversion procedure. We show that the BWT and the Alternating BWT are the only rank-invertible transformations in the class we have defined.