A Linear-Space Data Structure for Range-LCP Queries in Poly-Logarithmic Time

Abstract

Let $\mathsf {T}[1,n]$ be a text of length n and $\mathsf {T}[i,n]$ be the suffix starting at position i. Also, for any two strings X and Y, let $\mathsf {LCP}(X, Y)$ denote their longest common prefix. The range-LCP of $\mathsf {T}$ w.r.t. a range $[\alpha ,\beta ]$, where $1\le \alpha < \beta \le n$ is

Amir et al. [ISAAC 2011] introduced the indexing version of this problem, where the task is to build a data structure over $\mathsf {T}$, so that $\mathsf {rlcp}(\alpha ,\beta )$ for any query range $[\alpha ,\beta ]$ can be reported efficiently. They proposed an $O(n\log ^{1+\epsilon } n)$ space structure with query time $O(\log \log n)$, and a linear space (i.e., O(n) words) structure with query time $O(\delta \log \log n)$, where $\delta = \beta -\alpha +1$ is the length of the input range and $\epsilon > 0$ is an arbitrarily small constant. Later, Patil et al. [SPIRE 2013] proposed another linear space structure with an improved query time of $O(\sqrt{\delta }\log ^{\epsilon } \delta )$. This poses an interesting question, whether it is possible to answer $\mathsf {rlcp}(\cdot ,\cdot )$ queries in poly-logarithmic time using a linear space data structure. In this paper, we settle this question by presenting an O(n) space data structure with query time $O(\log ^{1+\epsilon } n)$ and construction time $O(n\log n)$.

A part of this work was done at NII Shonan Meeting No. 126: Computation over Compressed Structured Data.