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 $O(n^3 |N|^3 + |N|^4)$, where $n$ is the input length and $|N|$ is the number of non-terminals in the grammar. In contrast, our speed-up runs in $O(n^2 |N|^3+n^3|N|^2)$.

翻译：已知有多种算法能够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，良好算法的运行时间通常与输入字符串长度的立方成正比。然而，某些已有算法在文法的规模方面并非最优。本文提出了一种对Jelinek与Lafferty（1991）算法的新型加速方案，原算法的时间复杂度为$O(n^3 |N|^3 + |N|^4)$，其中$n$为输入长度，$|N|$为文法中的非终结符数量；而本文加速方案的时间复杂度为$O(n^2 |N|^3+n^3|N|^2)$。