Faster and simpler online/sliding rightmost Lempel-Ziv factorizations
Wataru Sumiyoshi, Takuya Mieno, Shunsuke Inenaga
TL;DR
This work addresses the online/sliding rightmost LZ-factorizations and LPF arrays, introducing BP-linked trees to maintain dynamic tree representations and enabling fast rightmost references. By integrating BP-linked suffix trees with enhanced dynamic RMQ/RMQ on a dynamic list, the authors achieve $O(n \log n / \log\log n)$ time for online/sliding rightmost LPF/LZ computations with $O(n)$ space (integer alphabets), and $O(n \log d / \log\log d)$ time with $O(d)$ space for sliding-window versions. The framework also yields efficient online computations for related factorizations, such as longest/minimum closed factorizations, via RMQ-based queries on the BP representation and reductions to dynamic subtree minimum queries. These results offer simpler, near-optimal data-structure-based solutions for rightmost LZ/LZ-like factorizations with practical implications for compression and string-processing tasks.
Abstract
We tackle the problems of computing the rightmost variant of the Lempel-Ziv factorizations in the online/sliding model. Previous best bounds for this problem are O(n log n) time with O(n) space, due to Amir et al. [IPL 2002] for the online model, and due to Larsson [CPM 2014] for the sliding model. In this paper, we present faster O(n log n/log log n)-time solutions to both of the online/sliding models. Our algorithms are built on a simple data structure named BP-linked trees, and on a slightly improved version of the range minimum/maximum query (RmQ/RMQ) data structure on a dynamic list of integers. We also present other applications of our algorithms.