Compressed Indexing for Consecutive Occurrences

The fundamental question considered in algorithms on strings is that of indexing, that is, preprocessing a given string for specific queries. By now we have a number of efficient solutions for this problem when the queries ask for an exact occurrence of a given pattern $P$. However, practical applications motivate the necessity of considering more complex queries, for example concerning near occurrences of two patterns. Recently, Bille et al. [CPM 2021] introduced a variant of such queries, called gapped consecutive occurrences, in which a query consists of two patterns $P_{1}$ and $P_{2}$ and a range $[a,b]$, and one must find all consecutive occurrences $(q_1,q_2)$ of $P_{1}$ and $P_{2}$ such that $q_2-q_1 \in [a,b]$. By their results, we cannot hope for a very efficient indexing structure for such queries, even if $a=0$ is fixed (although at the same time they provided a non-trivial upper bound). Motivated by this, we focus on a text given as a straight-line program (SLP) and design an index taking space polynomial in the size of the grammar that answers such queries in time optimal up to polylog factors.