Deterministic sub-linear space LCE data structures with efficient construction

TL;DR

This paper introduces a deterministic data structure for LCE queries that balances space and query time efficiently, while significantly reducing construction time compared to previous methods.

Contribution

It presents a new deterministic LCE data structure with optimal space, improved query time, and faster construction than prior solutions.

Findings

Achieves $O(n/ au)$ space and $O( au ext{min}\{ ext{log} au, ext{log}rac{n}{ au} ight)$ query time.

Construction time is reduced to $O(n au)$, significantly faster than previous quadratic solutions.

Provides a practical deterministic approach for efficient string processing applications.

Abstract

Given a string $S$ of $n$ symbols, a longest common extension query $\mathsf{LCE}(i,j)$ asks for the length of the longest common prefix of the $i$th and $j$th suffixes of $S$. LCE queries have several important applications in string processing, perhaps most notably to suffix sorting. Recently, Bille et al. (J. Discrete Algorithms 25:42-50, 2014, Proc. CPM 2015: 65-76) described several data structures for answering LCE queries that offers a space-time trade-off between data structure size and query time. In particular, for a parameter $1 \leq \tau \leq n$, their best deterministic solution is a data structure of size $O(n/\tau)$ which allows LCE queries to be answered in $O(\tau)$ time. However, the construction time for all deterministic versions of their data structure is quadratic in $n$. In this paper, we propose a deterministic solution that achieves a similar space-time trade-off of $O(\tau\min\{\log\tau,\log\frac{n}{\tau}\})$ query time using $O(n/\tau)$ space, but significantly improve the construction time to $O(n\tau)$.