  The LCP array is an important tool in stringology, allowing to speed up
pattern matching algorithms and enabling compact representations of the suffix
tree. Recently, Conte et al. [DCC 2023] and Cotumaccio et al. [SPIRE 2023]
extended the definition of this array to Wheeler DFAs and, ultimately, to
arbitrary labeled graphs, proving that it can be used to efficiently solve
matching statistics queries on the graph's paths. In this paper, we provide the
first efficient algorithm building the LCP array of a directed labeled graph
with $n$ nodes and $m$ edges labeled over an alphabet of size $\sigma$. After
arguing that the natural generalization of a compact-space LCP-construction
algorithm by Beller et al. [J. Discrete Algorithms 2013] runs in time
$\Omega(n\sigma)$, we present a new algorithm based on dynamic range stabbing
building the LCP array in $O(n\log \sigma)$ time and $O(n\log\sigma)$ bits of
working space.
