Clustering, order conditions, and languages of interval exchanges

TL;DR

This paper explores the relationship between clustering in the Burrows-Wheeler transform and languages of interval exchange transformations, revealing new characterizations of return words and palindromic properties.

Contribution

It establishes a connection between clustering of words and their role as return words in interval exchange transformations, answering a specific open question and analyzing palindromic richness.

Findings

Primitive words cluster iff they are return words in generalized interval exchange transformations.

Symmetric transformations produce palindromically rich languages.

Orders of induced transformations match original orders iff shortest bispecial words are palindromes.

Abstract

We investigate various connections between the clustering for the Burrows-Wheeler transform, a lossless algorithm used in data compression, and languages of interval exchange transformations. We show that a primitive word $u$ clusters for a pair of orders $(<_D,<_A)$ if and only if $u$ is a return word in the natural coding of a generalised interval exchange transformation with departure and arrival orders $(<_D,<_A)$. This answers a question of M. Lapointe on the perfect clustering of return words for a symmetric standard interval exchange transformation. We show that if $T$ is symmetric, then all natural codings are palindromically rich languages, and the orders of the induced transformation on a cylinder $[w]$ equal the original departure/arrival orders $(<_D,<_A)$ if and only if the shortest bispecial word containing $w$ is a palindrome. We also investigate language related features distinguishing between standard and generalised interval exchange transformations.