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 $\mathcal{O}({N^2 |\mathcal{N}|^3+N^3|\mathcal{N}|^2})$.

翻译：已知有多个算法能高效计算概率上下文无关文法(PCFG)下字符串的前缀概率。针对该问题的优秀算法的时间复杂度为输入字符串长度的立方级。然而，部分现有算法在文法规模方面并非最优。本文提出Jelinek和Lafferty(1991)算法的一种新颖加速方案，其原始算法运行时间为$\mathcal{O}({N^3 |\mathcal{N}|^3 + |\mathcal{N}|^4})$（其中$N$为输入长度，$|\mathcal{N}|$为文法中非终结符数量），而本文加速版本的运行时间降至$\mathcal{O}({N^2 |\mathcal{N}|^3+N^3|\mathcal{N}|^2})$。