A Fast Algorithm for Computing Prefix Probabilities

Multiple algorithms are known for efficiently calculating the prefix probability of a string under a probabilistic context-free grammar (PCFG). Good algorithms for the problem have a runtime cubic in the length of the input string. However, some proposed algorithms are suboptimal with respect to the size of the grammar. This paper proposes a novel speed-up of Jelinek and Lafferty's (1991) algorithm, which runs in $\mathcal{O}({N^3 |\mathcal{N}|^3 + |\mathcal{N}|^4})$, where $N$ is the input length and $|\mathcal{N}|$ is the number of non-terminals in the grammar. In contrast, our speed-up runs in $\mathcal{O}({N^2 |\mathcal{N}|^3+N^3|\mathcal{N}|^2})$.