Efficient Exact Paths For Dyck and semi-Dyck Labeled Path Reachability

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 $\widetilde{O}(n^ω)$ exact path solution. Here $ω$ 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 $\widetilde{O}(n^ω)$. 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.