Graphs can be succinctly indexed for pattern matching in $ O(|E|^2 + |V|^{5 / 2}) $ time

For the first time we provide a succinct pattern matching index for arbitrary graphs that can be built in polynomial time, which requires less space and answers queries more efficiently than the one in [SODA 2021]. We show that, given an edge-labeled graph $G = (V, E)$, there exists a data structures of $|E /_{\le_G}|(\lceil \log|\Sigma|\rceil + \lceil\log q\rceil + 2)\cdot (1+o(1)) + |V /_{\le_G}|\cdot (1+o(1))$ bits which can be built in $O(|E|^2 + |V /_{\le_G}|^{5 / 2})$ time and supports pattern matching on $G$ in $O(|P| \cdot q^2 \cdot \log(q\cdot |\Sigma|))$ time, where $G /_{\le_G} = (V /_{\le_G}, E /_{\le_G})$ is a quotient graph obtained by collapsing some nodes in $G$ (so $|V /_{\le_G}| \le |V|$ and $|E /_{\le_G}| \le |E|$) and $q$ is the width of the maximum co-lex relation on $G$. Our results have relevant applications in automata theory. First, we can build a succinct data structure to decide whether a string is accepted by a given automaton. Second, starting from an automaton $\mathcal{A}$, one can define a relation $\preceq_\mathcal{A}$ and a quotient automaton that capture the nondeterminism of $\mathcal{A}$, improving the results in [SODA 2021].