Online LZ77 Parsing and Matching Statistics with RLBWTs

Lempel-Ziv 1977 (LZ77) parsing, matching statistics and the Burrows-Wheeler Transform (BWT) are all fundamental elements of stringology. In a series of recent papers, Policriti and Prezza (DCC 2016 and Algorithmica, CPM 2017) showed how we can use an augmented run-length compressed BWT (RLBWT) of the reverse $T^R$ of a text $T$, to compute offline the LZ77 parse of $T$ in $O (n \log r)$ time and $O (r)$ space, where $n$ is the length of $T$ and $r$ is the number of runs in the BWT of $T^R$. In this paper we first extend a well-known technique for updating an unaugmented RLBWT when a character is prepended to a text, to work with Policriti and Prezza's augmented RLBWT. This immediately implies that we can build online the LZ77 parse of $T$ while still using $O (n \log r)$ time and $O (r)$ space; it also seems likely to be of independent interest. Our experiments, using an extension of Ohno, Takabatake, I and Sakamoto's (IWOCA 2017) implementation of updating, show our approach is both time- and space-efficient for repetitive strings. We then show how to augment the RLBWT further --- albeit making it static again and increasing its space by a factor proportional to the size of the alphabet --- such that later, given another string $S$ and $O (\log \log n)$-time random access to $T$, we can compute the matching statistics of $S$ with respect to $T$ in $O (|S| \log \log n)$ time.