Online Computation of the Longest Repeating Suffix and Smallest Suffixient Sets via Incremental Run-Length BWT-based Indexes

2026-07-06

AI Summary

This study addresses the online computation of the Longest Repeated Suffix (LRS) and the maintenance of a minimal suffix-dense set. The authors propose a novel approach based on an incremental run-length compressed Burrows–Wheeler Transform (BWT) index, enabling efficient updates of both structures within compressed space. Two time–space trade-offs are presented, achieving per-character processing times of \(O((\log n / \log \log n)^2)\) while using only \(O(n)\) or \(O(n \log \log n)\) bits of working space on highly repetitive texts—outperforming existing methods. Notably, this work achieves the first online construction of a minimal suffix-dense set in compressed space, provides worst-case time guarantees for LRS computation, and establishes a space lower bound for deterministic online LRS algorithms.

Abstract

We revisit the online construction of \emph{smallest suffixient sets} and the online computation of the \emph{longest repeating suffix} (LRS). We give the first compressed-space online construction of smallest suffixient sets, and present two space-time trade-offs for both problems: $O(r\log n+n)$ bits of working space and $O(\log^2 n/\log \log n)$ worst-case time per character, and $O(r\log n+n \log \log n)$ bits of working space and $O((\log n/\log \log n)^2)$ worst-case time per character. Here, $r$ is the number of runs in the Burrows-Wheeler transform of the reverse of $T[1..n]$. In particular, for highly repetitive texts satisfying $r=O(n/\log n)$, the first trade-off uses $O(n)$ bits of working space, while the second uses $O(n\log\log n)$ bits. We also prove that any deterministic online algorithm for computing LRS requires \(Ω(n)\) bits of peak working space in the worst case, even over a constant-size alphabet. Through reductions from online LRS computation, we extend this lower bound to deterministic online algorithms maintaining either an arbitrary smallest suffixient set augmented with the length of the supermaximal right extension represented by each selected position, or the position-only smallest suffixient set obtained by selecting the rightmost occurrence of every such extension. For constructing smallest suffixient sets, our algorithms are the first online solutions using compressed working space, improving the $O(n)$-word space required by previous online constructions. For compressed-space online LRS computation, compared with the algorithm of Prezza and Rosone~[CiE 2020], our bounds improve their $O(\log^2 n)$ amortized time per character by factors of $Θ(\log\log n)$ and $Θ((\log\log n)^2)$, respectively, while also providing worst-case guarantees.

Problem

• longest repeating suffix
• smallest suffixient sets
• online computation
• compressed space
• Burrows-Wheeler transform

Innovation

• online algorithm
• compressed space
• Burrows-Wheeler Transform
• longest repeating suffix
• smallest suffixient set