On Arithmetically Progressed Suffix Arrays and related Burrows-Wheeler Transforms

TL;DR

This paper characterizes strings with suffix arrays based on arithmetic progressions, linking them to specific classes of words and analyzing their Burrows-Wheeler transforms, thus advancing understanding of string structure and transformations.

Contribution

It provides a complete characterization of arithmetically progressed suffix arrays and their associated strings, including connections to well-known word classes and the shape of their Burrows-Wheeler transforms.

Findings

Suffix arrays with arithmetic progressions correspond to unary, binary, or ternary strings.

Binary case links to Christoffel, balanced, and Fibonacci words.

The shape of the Burrows-Wheeler transform for these strings is described.

Abstract

We characterize those strings whose suffix arrays are based on arithmetic progressions, in particular, arithmetically progressed permutations where all pairs of successive entries of the permutation have the same difference modulo the respective string length. We show that an arithmetically progressed permutation $P$ coincides with the suffix array of a unary, binary, or ternary string. We further analyze the conditions of a given $P$ under which we can find a uniquely defined string over either a binary or ternary alphabet having $P$ as its suffix array. For the binary case, we show its connection to lower Christoffel words, balanced words, and Fibonacci words. In addition to solving the arithmetically progressed suffix array problem, we give the shape of the Burrows-Wheeler transform of those strings solving this problem. These results give rise to numerous future research directions.