Suffixient Sets

We define a suffixient set for a text $T [1..n]$ to be a set $S$ of positions between 1 and $n$ such that, for any edge descending from a node $u$ to a node $v$ in the suffix tree of $T$, there is an element $s \in S$ such that $u$'s path label is a suffix of $T [1..s - 1]$ and $T [s]$ is the first character of $(u, v)$'s edge label. We first show there is a suffixient set of cardinality at most $2 \bar{r}$, where $\bar{r}$ is the number of runs in the Burrows-Wheeler Transform of the reverse of $T$. We then show that, given a straight-line program for $T$ with $g$ rules, we can build an $O (\bar{r} + g)$-space index with which, given a pattern $P [1..m]$, we can find the maximal exact matches (MEMs) of $P$ with respect to $T$ in $O (m \log (\sigma) / \log n + d \log n)$ time, where $\sigma$ is the size of the alphabet and $d$ is the number of times we would fully or partially descend edges in the suffix tree of $T$ while finding those MEMs.