# The Alternating BWT: an algorithmic perspective

## TL;DR

This paper provides a comprehensive combinatorial and algorithmic analysis of the Alternating Burrows-Wheeler Transform (ABWT), demonstrating its invertibility, efficient computation, and applicability in compressed full-text indexing, analogous to the classical BWT.

## Contribution

It establishes the invertibility and efficient computation of ABWT, extending BWT techniques to an alternating lexicographical order and enabling its use in compressed indexing.

## Key findings

- ABWT is rank-invertible, like BWT.
- Efficient algorithms for ABWT computation are developed.
- ABWT can be used for compressed full-text indices.

## Abstract

The Burrows-Wheeler Transform (BWT) is a word transformation introduced in 1994 for Data Compression. It has become a fundamental tool for designing self-indexing data structures, with important applications in several area in science and engineering. The Alternating Burrows-Wheeler Transform (ABWT) is another transformation recently introduced in [Gessel et al. 2012] and studied in the field of Combinatorics on Words. It is analogous to the BWT, except that it uses an alternating lexicographical order instead of the usual one. Building on results in [Giancarlo et al. 2018], where we have shown that BWT and ABWT are part of a larger class of reversible transformations, here we provide a combinatorial and algorithmic study of the novel transform ABWT. We establish a deep analogy between BWT and ABWT by proving they are the only ones in the above mentioned class to be rank-invertible, a novel notion guaranteeing efficient invertibility. In addition, we show that the backward-search procedure can be efficiently generalized to the ABWT; this result implies that also the ABWT can be used as a basis for efficient compressed full text indices. Finally, we prove that the ABWT can be efficiently computed by using a combination of the Difference Cover suffix sorting algorithm [K\"{a}rkk\"{a}inen et al., 2006] with a linear time algorithm for finding the minimal cyclic rotation of a word with respect to the alternating lexicographical order.