Logarithmic equal-letter runs for BWT of purely morphic words

TL;DR

This paper investigates the number of equal-letter runs in the BWT of purely morphic words, establishing bounds that relate to the words' structure and providing insights into BWT efficiency for these words.

Contribution

It proves that the BWT of purely morphic binary words has a logarithmic number of runs, with new structural properties of bispecial circular factors underpinning these bounds.

Findings

For binary purely morphic words, r_bwt = O(log n).

For many binary morphic words, r_bwt = Θ(log n).

Structural properties of bispecial circular factors are key to these bounds.

Abstract

In this paper we study the number $r_{bwt}$ of equal-letter runs produced by the Burrows-Wheeler transform ($BWT$) when it is applied to purely morphic finite words, which are words generated by iterating prolongable morphisms. Such a parameter $r_{bwt}$ is very significant since it provides a measure of the performances of the $BWT$, in terms of both compressibility and indexing. In particular, we prove that, when $BWT$ is applied to any purely morphic finite word on a binary alphabet, $r_{bwt}$ is $\mathcal{O}(\log n)$, where $n$ is the length of the word. Moreover, we prove that $r_{bwt}$ is $\Theta(\log n)$ for the binary words generated by a large class of prolongable binary morphisms. These bounds are proved by providing some new structural properties of the \emph{bispecial circular factors} of such words.