  Until recently, most experts would probably have agreed we cannot
backwards-step in constant time with a run-length compressed Burrows-Wheeler
Transform (RLBWT), since doing so relies on rank queries on sparse bitvectors
and those inherit lower bounds from predecessor queries. At ICALP '21, however,
Nishimoto and Tabei described a new, simple and constant-time implementation.
For a permutation $\pi$, it stores an $O (r)$-space table -- where $r$ is the
number of positions $i$ where either $i = 0$ or $\pi (i + 1) \neq \pi (i) + 1$
-- that enables the computation of successive values of $\pi(i)$ by table
look-ups and linear scans. Nishimoto and Tabei showed how to increase the
number of rows in the table to bound the length of the linear scans such that
the query time for computing $\pi(i)$ is constant while maintaining $O
(r)$-space.
  In this paper we refine Nishimoto and Tabei's approach, including a
time-space tradeoff, and experimentally evaluate different implementations
demonstrating the practicality of part of their result. We show that even
without adding rows to the table, in practice we almost always scan only a few
entries during queries. We propose a decomposition scheme of the permutation
$\pi$ corresponding to the LF-mapping that allows an improved compression of
the data structure, while limiting the query time. We tested our implementation
on real-world genomic datasets and found that without compression of the table,
backward-stepping is drastically faster than with sparse bitvector
implementations but, unfortunately, also uses drastically more space. After
compression, backward-stepping is competitive both in time and space with the
best existing implementations.
