Gapped Indexing for Consecutive Occurrences

Abstract

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