Gapped Indexing for Consecutive Occurrences

The classic string indexing problem is to preprocess a S into compact data structure that supports efficient pattern matching queries. Typical queries include existential (decide if the occurs in S), reporting (return all positions where occurs), and counting number of occurrences pattern). In this paper we consider variant indexing, goal compactly represent such given two patterns $$P_1$$ $$P_2$$ gap range $${[}\alpha , \beta ]$$ can quickly find consecutive with distance i.e., pairs subsequent within range. We present structures use linear space query time $${\widetilde{O}}(|P_1|+|P_2|+n^{2/3})$$ for existence $${\widetilde{O}}(|P_1|+|P_2|+n^{2/3}\hbox {occ}^{1/3})$$ reporting. complement conditional lower bound based on set intersection showing any solution using $${\widetilde{O}}(n)$$ must $${\widetilde{\Omega }}(|P_1| + |P_2| \sqrt{n})$$ time. To obtain our results develop new techniques ideas independent interest including suffix tree decomposition hardness problem.