Language Edit Distance & Maximum Likelihood Parsing of Stochastic Grammars: Faster Algorithms & Connection to Fundamental Graph Problems

Given a context free language $L(G)$ over alphabet $\Sigma$ and a string $s \in \Sigma^*$, the language edit distance (Lan-ED) problem seeks the minimum number of edits (insertions, deletions and substitutions) required to convert $s$ into a valid member of $L(G)$. The well-known dynamic programming algorithm solves this problem in $O(n^3)$ time (ignoring grammar size) where $n$ is the string length [Aho, Peterson 1972, Myers 1985]. Despite its vast number of applications, there is no algorithm known till date that computes or approximates Lan-ED in true sub-cubic time. In this paper we give the first such algorithm that computes Lan-ED almost optimally. For any arbitrary $\epsilon > 0$, our algorithm runs in $\tilde{O}(\frac{n^{{\omega}}{poly(\epsilon)})$} time and returns an estimate within a multiplicative approximation factor of $(1+\epsilon)$, where $\omega$ is the exponent of ordinary matrix multiplication of $n$ dimensional square matrices. It also computes the edit script. Further, for all substrings of $s$, we can estimate their Lan-ED within $(1\pm \epsilon)$ factor in $\tilde{O}(\frac{n^{{\omega}}{poly(\epsilon)})$} time with high probability. We also design the very first sub-cubic ($\tilde{O}(n^\omega)$) algorithm to handle arbitrary stochastic context free grammar (SCFG) parsing. SCFGs lie at the foundation of statistical natural language processing, they generalize hidden Markov models, and have found widespread applications. To complement our upper bound result, we show that exact computation of SCFG parsing, or Lan-ED with insertion as only edit operation in true sub-cubic time will imply a truly sub-cubic algorithm for all-pairs shortest paths, and hence to a large range of problems in graphs and matrices. Known lower bound results on parsing implies no improvement over our time bound of $O(n^\omega)$ is possible for any nontrivial multiplicative approximation.