  String kernels are typically used to compare genome-scale sequences whose
length makes alignment impractical, yet their computation is based on data
structures that are either space-inefficient, or incur large slowdowns. We show
that a number of exact string kernels, like the $k$-mer kernel, the substrings
kernels, a number of length-weighted kernels, the minimal absent words kernel,
and kernels with Markovian corrections, can all be computed in $O(nd)$ time and
in $o(n)$ bits of space in addition to the input, using just a
$\mathtt{rangeDistinct}$ data structure on the Burrows-Wheeler transform of the
input strings, which takes $O(d)$ time per element in its output. The same
bounds hold for a number of measures of compositional complexity based on
multiple value of $k$, like the $k$-mer profile and the $k$-th order empirical
entropy, and for calibrating the value of $k$ using the data.
