  The compact directed acyclic word graph (CDAWG) of a string $T$ of length $n$
takes space proportional just to the number $e$ of right extensions of the
maximal repeats of $T$, and it is thus an appealing index for highly repetitive
datasets, like collections of genomes from similar species, in which $e$ grows
significantly more slowly than $n$. We reduce from $O(m\log{\log{n}})$ to
$O(m)$ the time needed to count the number of occurrences of a pattern of
length $m$, using an existing data structure that takes an amount of space
proportional to the size of the CDAWG. This implies a reduction from
$O(m\log{\log{n}}+\mathtt{occ})$ to $O(m+\mathtt{occ})$ in the time needed to
locate all the $\mathtt{occ}$ occurrences of the pattern. We also reduce from
$O(k\log{\log{n}})$ to $O(k)$ the time needed to read the $k$ characters of the
label of an edge of the suffix tree of $T$, and we reduce from
$O(m\log{\log{n}})$ to $O(m)$ the time needed to compute the matching
statistics between a query of length $m$ and $T$, using an existing
representation of the suffix tree based on the CDAWG. All such improvements
derive from extracting the label of a vertex or of an arc of the CDAWG using a
straight-line program induced by the reversed CDAWG.
