Sparse Regular Expression Matching

We present the first algorithm for regular expression matching that can take advantage of sparsity in the input instance. Our main result is a new algorithm that solves regular expression matching in $O\left(\Delta \log \log \frac{nm}{\Delta} + n + m\right)$ time, where $m$ is the number of positions in the regular expression, $n$ is the length of the string, and $\Delta$ is the \emph{density} of the instance, defined as the total number of states in a simulation of the position automaton. This measure is a lower bound on the total number of states in simulations of all classic polynomial sized finite automata. Our bound improves the best known bounds for regular expression matching by almost a linear factor in the density of the problem. The key component in the result is a novel linear space representation of the position automaton that supports state-set transition computation in near-linear time in the size of the input and output state sets.