On the number of equal-letter runs of the bijective Burrows-Wheeler transform

The Bijective Burrows-Wheeler Transform (BBWT) is a variant of the famous BWT [Burrows and Wheeler, 1994]. The BBWT was introduced by Gil and Scott in 2012, and is based on the extended BWT of Mantaci et al. [TCS 2007] and on the Lyndon factorization of the input string. In the original paper, the compression achieved with the BBWT was shown to be competitive with that of the BWT, and it has been gaining interest in recent years. In this work, we present the first study of the number $r_B$ of runs of the BBWT, which is a measure of its compression power. We exhibit an infinite family of strings on which $r_B$ of the string and of its reverse differ by a multiplicative factor of $\Theta (\log n)$, where n is the length of the string. We also give several theoretical results on the BBWT, including a characterization of binary strings for which the BBWT has two runs. Finally, we present experimental results and statistics on $r_B\left(s\right)$ and $r_B\left(s^{\text{rev}}\right)$, as well as on the number of Lyndon factors in the Lyndon factorization of s and $s^{\text{rev}}$.