Range LCP Queries Revisited

Abstract

The Range LCP problem is to preprocess a string $S[1\dots n]$, to enable efficient solutions of the following query: given a range [l, r] as the input, report $\max _{i, j \in \{l,\ldots ,r\}} |\mathsf {LCP}(S_{i}, S_j)|$. Here $\mathsf {LCP}(S_i, S_j)$ is the longest common prefix of the suffixes of S starting at locations i and j and $|\mathsf {LCP}(S_i,S_j)|$ is its length. We study a natural extension of this problem, where the query consists of two ranges. Additionally, we allow a bounded number (say $k\ge 0$) of mismatches in the $\mathsf {LCP}$ computation. Specifically, our task is to report the following when two ranges $[\ell _1, r_1]$ and $[\ell _2,r_2]$ comes as input:

$$\max _{\{\ell _1\le i\le r_1, \ell _2\le j\le r_2\}}|\mathsf {LCP}_k(S_i,S_j)|$$

Here $\mathsf {LCP}_k(S_i,S_j)$ is the longest prefix of $S_i$ and $S_j$ with at most k mismatches allowed. We show that the queries can be answered in O(k) time using an $O(n^2/w)$ space data structure, where w is the word size. We also present space efficient data structures for $k=0$ and $k=1$. For $k=0$, we obtain a linear space data structure with query time $O(\sqrt{n/w}\log ^{\epsilon } n)$, where w is the word size and $\epsilon >0$ is an arbitrarily small constant.

For the case $k=1$ we obtain an $O(n\log n)$ space data structure with query time $O(\sqrt{n}\log n)$.

Finally, we give a reduction from Set Intersection to Range LCP queries, suggesting that it will be very difficult to improve our upper bound by more than a factor of $O(\log ^{\epsilon }n)$.

A. Amir—Partly supported by ISF grant 571/14.

M. Lewenstein—Partly supported by GIF 1147/2011 and BSF 2010437.