Let $G = (V, E)$ be a digraph where each vertex is unlabeled, each edge is labeled by a character in some alphabet $\Omega$, and any two edges with both the same head and the same tail have different labels. The powerset construction gives a transform of $G$ into a weakly connected digraph $G' = (V', E')$ that enables solving the decision problem of whether there is a walk in $G$ matching an arbitrarily long query string $q$ in time linear in $|q|$ and independent of $|E|$ and $|V|$. We show $G$ is uniquely determined by $G'$ when for every $v_\ell \in V$, there is some distinct string $s_\ell$ on $\Omega$ such that $v_\ell$ is the origin of a closed walk in $G$ matching $s_\ell$, and no other walk in $G$ matches $s_\ell$ unless it starts and ends at $v_\ell$. We then exploit this invertibility condition to strategically alter any $G$ so its transform $G'$ enables retrieval of all $t$ terminal vertices of walks in the unaltered $G$ matching $q$ in $O(|q| + t \log |V|)$ time. We conclude by proposing two defining properties of a class of transforms that includes the Burrows-Wheeler transform and the transform presented here.
翻译:让 $G = ( V, E) 成为一条分解线, 其中每个顶端都不贴标签, 每个边缘都标有某种字母的字符标签, 以美元和美元为单位, 而任何两边的首端和尾端都有不同的标签。 电源设置构造将G $ 转换成一个连接薄弱的下划字 $G = ( V, E), 从而能够解决一个决定问题, 即是否用G $ 匹配任意长的查询字符串 $, 时间直线, 以 $ $ $ 计, 且不以 $ 美元 和 $ 美元为单位。 我们显示, 当每张美元 + 美元 和 美元 双边边边的边的边的边的边际, 包括 美元 折叠 。 我们随后在平整页的折价 中 将 美元 变换成任何战略性 G 。