Efficient exact paths for dyck and semi-dyck labeled path reachability (extended abstract)

Consider any two vertices in a weighted digraph. The exact path length problem is to determine if there is a path of a given fixed cost between these vertices. This paper focuses on the exact path problem for costs −1,0 or +1 between all pairs of vertices. This special case is also restricted to original edge weights from {−1, +1}. In this special case, this paper gives an O(nω log2 n) exact path solution, where ω is the best exponent for matrix multiplication. Basic variations of this algorithm determine which pairs of digraph nodes have Dyck or semi-Dyck labeled paths between them, assuming two terminals or parenthesis. Therefore, determining reachability for Dyck or semi-Dyck labeled paths costs O(nω log2 n). Both the exact path and labeled path solutions can be improved by poly-log factors, but these improvements are not given here. To find Dyck or semi-Dyck reachability in a labeled digraph, each edge has a single symbol on it. A path label is made by concatenating all symbols along the path. Cycles are allowed without any repeated edges. These paths have the same number of balanced parenthesizations (semi-Dyck languages) or have an equal numbers of matching symbols (Dyck languages). The exact path length problem has many applications. These applications include the labeled path problems given here, which in turn, have many applications.