Succinct Non-overlapping Indexing

Abstract

Text indexing is a fundamental problem in computer science. The objective is to preprocess a text T, so that, given a pattern P, we can find all starting positions (or simply, occurrences) of P in $T$ efficiently. In some cases, additional restrictions are imposed. We consider two variants, namely the non-overlapping indexing problem, and the range non-overlapping indexing problem. Given a text $T$ having n characters, the non-overlapping indexing problem is defined as follows: pre-process $T$ into a data structure, such that for any pattern P, containing |P| characters, we can report a set containing the maximum number of non-overlapping occurrences of P in $T$. Cohen and Porat (in: Algorithms and computation, 20th international symposium, ISAAC 2009, Honolulu, Hawaii. Proceedings, 2009) showed that by maintaining a linear space index in which the suffix tree of $T$ is augmented with an O(n) word data structure, a query P can be answered in optimal time $O(|P|+nocc)$, where $nocc$ is the number of occurrences reported. We present the following new result. Let $\mathsf {CSA}$ (not necessarily a compressed suffix array) be an index of $T$ that can compute (i) the suffix range of P in $\mathsf {search}(P)$ time, and (ii) a suffix array or an inverse suffix array value in $\mathsf {t}_\mathsf {SA}$ time. By using $\mathsf {CSA}$ alone, we can answer a query P in $\mathsf {search}(P)+\mathsf {sort}(nocc)+O(nocc\cdot \mathsf {t}_\mathsf {SA})$ time. The function $\mathsf {sort}(k)$ denotes the time for sorting k numbers in $\{1,2,\dots ,n\}$. In the range non-overlapping indexing problem, along with the pattern P, two integers a and b, $b \ge a$, are provided as input. The task is to report a set containing the maximum number of non-overlapping occurrences of P that lie within the range [a, b]. For any arbitrarily small positive constant $\epsilon$, we present an $O(n \log ^\epsilon n)$ word index with $O(|P| + nocc_{a,b})$ query time, where $nocc_{a,b}$ is the number of occurrences reported. Our index improves upon the result of Cohen and Porat [6].