An Efficient Data Structure and Algorithm for Long-Match Query in Run-Length Compressed BWT

String matching problems in bioinformatics are typically for finding exact substring matches between a query and a reference text. Previous formulations often focus on maximum exact matches (MEMs). However, multiple occurrences of substrings of the query in the text that are long enough but not maximal may not be captured by MEMs. Such long matches can be informative, especially when the text is a collection of similar sequences such as genomes. In this paper, we describe a new type of match between a pattern and a text that aren't necessarily maximal in the query, but still contain useful matching information: locally maximal exact matches (LEMs). There are usually a large amount of LEMs, so we only consider those above some length threshold $ℒ$ . These are referred to as long LEMs. The purpose of long LEMs is to capture substring matches between a query and a text that are not necessarily maximal in the pattern but still long enough to be important. Therefore efficient long LEMs finding algorithms are desired for these datasets. However, these datasets are too large to query on traditional string indexes. Fortunately, these datasets are very repetitive. Recently, compressed string indexes that take advantage of the redundancy in the data but retain efficient querying capability have been proposed as a solution. We therefore give an efficient algorithm for computing all the long LEMs of a query and a text in a BWT runs compressed string index. We describe an $O(m+occ)$ expected time algorithm that relies on an $O\left(r\right)$ words space string index for outputting all long LEMs of a pattern with respect to a text given the matching statistics of the pattern with respect to the text. Here $m$ is the length of the query, $occ$ is the number of long LEMs outputted, and $r$ is the number of runs in the BWT of the text. The $O\left(r\right)$ space string index we describe relies on an adaptation of the move data structure by Nishimoto and Tabei. We are able to support $LCP\left[i\right]$ queries in constant time given $SA\left[i\right]$ . In other words, we answer $\text{PLCP}\left[i\right]$ queries in constant time. These $\text{PLCP}$ queries enable the efficient long LEM query. Long LEMs may provide useful similarity information between a pattern and a text that MEMs may ignore. This information is particularly useful in pangenome and biobank scale haplotype panel contexts.