Burrows-Wheeler Transform of Words Defined by Morphisms

Abstract

The Burrows-Wheeler transform (BWT) is a popular method used for text compression. It was proved that BWT has optimal performance on standard words, i.e. the building blocks of Sturmian words. In this paper, we study the application of BWT on more general morphic words: the Thue-Morse word and to generalizations of the Fibonacci word to alphabets with more than two letters; then, we study morphisms obtained as composition of the Thue-Morse morphism with a Sturmian one. In all these cases, the BWT efficiently clusters the iterates of the morphisms generating prefixes of these infinite words, for which we determine the compression clustering ratio.

On repetitiveness measures of Thue-Morse words

“…We note that for any standard Sturmian word s, z(s) = Θ(log |s|) [1], while the size r(s) of the RLBWT is always constant [14]. On the other hand, z(t n ) and r(t n ) are both Θ(n), i.e., logarithmic in the length |t n | (the former due to [1] as well as this work, and the latter due to [3]). This shows that Thue-Morse words are an example where the size of smallest string attractor is not a tight lower bound on the size of the smallest of the known efficiently computable dictionary compressed representation, namely, min{z(w), r(w)}.…”

Logarithmic equal-letter runs for BWT of purely morphic words

“…Note that such upper bounds extend some known results. In fact, as shown in [1], r bwt (τ i (a)) = Θ(i). We also remark that, since n = |τ i (a)| = 2 i , r bwt (τ i (a)) = Θ(log n).…”

“…We also remark that, since n = |τ i (a)| = 2 i , r bwt (τ i (a)) = Θ(log n). Furthermore, from results provided in [1] and [18], it can be deduced that τ and θ are BW T -highly compressible. However, the lower bounds can be quite different.…”

“…the words ϕ i (a) generated by iterating a morphism ϕ prolongable on a. In [1], the parameter ρ has been computed for the families of finite words generated by some morphisms. In all cases considered in [1], the BW T efficiently clusters since the value of ρ is much lower than 1.…”

“…In [1], the parameter ρ has been computed for the families of finite words generated by some morphisms. In all cases considered in [1], the BW T efficiently clusters since the value of ρ is much lower than 1. In the one-page abstract appeared in [10] we extended such results by providing some new upper bounds on r bwt (ϕ i (a)) depending on the factor complexity of the fixed point ϕ ∞ (a).…”