How is CFL-reachability solvable in exponential time and space? I have read a paper which mentions that CFL-reachability is solvable in exponential time and space. Intuitively, I suppose that one need to explore through all the sub-paths in the PDA for a CFL. However, I am not able to convince myself with this explanation. If someone could provide an explanation, it would be great.
Please refer to the paper Program Analysis via Graph Reachability by
Thomas Reps where he mentions clearly that CF-reachability is solvable in cubic time.

I assume CFL-reachability is defined as follows:

A CFL-reachability problem is  where a path is considered to connect two nodes only if the concatenation of the labels on the edges of the path is a word in a particular context-free language (an $L$-path):

Definition 2.1 (from paper above). Let $L$ be a context-free language over alphabet $\Sigma$, and let $G$ be a graph whose edges are
labeled with members of $\Sigma$. Each path in $G$ defines a word over $\Sigma$, namely, the word obtained by concatenating, in order, the labels of the edges on the path. A path in $G$ is an $L$-path if its word is a member of
$L$.