  The exact path length problem is to determine if there is a path of a given
fixed cost between two vertices. This paper focuses on the exact path problem
for costs $-1,0$ or $+1$ between all pairs of vertices in an edge-weighted
digraph. The edge weights are from $\{ -1, +1 \}$. In this case, this paper
gives an $\widetilde{O}(n^{\omega})$ exact path solution. Here $\omega$ is the
best exponent for matrix multiplication and $\widetilde{O}$ is the asymptotic
upper-bound mod polylog factors.
  Variations of this algorithm determine which pairs of digraph nodes have Dyck
or semi-Dyck labeled paths between them, assuming two parenthesis. Therefore,
determining digraph reachability for Dyck or semi-Dyck labeled paths costs
$\widetilde{O}(n^{\omega})$. A path label is made by concatenating all symbols
along the path's edges.
  The exact path length problem has many applications. These applications
include the labeled path problems given here, which in turn, also have numerous
applications.
