Parse indexing for discarding short pseudo-MEMs safely

Brown et al.\ (2025) described a pre-processing step, called $k$-mer based breaking (KeBaB), that speeds up searching for long maximal exact matches (MEMs) between a pattern $P$ and an indexed repetitive text $T$. KeBaB produces a set of substrings of $P$ called pseudo-MEMs that often have total length much less than $\mathrm{\mid }P\mathrm{\mid }$ but are still guaranteed to contain all the MEMs of length at least a fixed parameter $k$. Brown et al.\ found that KeBaB can be particularly effective when we discard all but the longest pseudo-MEMs – but then we risk also discarding the longest MEMs! In this paper we show how we can use parse indexing to generate pseudo-MEMs together with lower bounds on the lengths of the longest MEMs they must contain, allowing us to discard short pseudo-MEMs safely.