Faster Lightweight Lempel-Ziv Parsing

Abstract

Abstract. We present an algorithm that computes the Lempel-Ziv decomposition in O(n(log σ + log log n)) time and n log σ + ǫn bits of space, where ǫ is a constant rational parameter, n is the length of the input string, and σ is the alphabet size. The n log σ bits in the space bound are for the input string itself which is treated as read-only.

Range Predecessor and Lempel-Ziv Parsing

“…All these approaches take in O(n log n) time. Very recently a solution with n log σ + O(n) space and O(n(log σ + log log n)) time has been proposed in [29]. We show a significant improvement -to O(n log log σ) time -is possible in the compact setting.…”

Refining the r-index

“…We implemented in C++ the online LZ77 parsing algorithm of Theorem 6 (the source code is available at [22]). There are lots of work for LZ77 parsing (e.g., see [31][32][33][34][35][36][37][38][39][40][41][42][43] and references therein). Among them we choose the ones whose implementations potentially work in the peak RAM usage smaller than n lg σ + n lg n bits and compare with our method.…”

Re-Pair in Small Space

“…A trade-off was proposed by Kärkkäinen et al [19], who needed O(n lg n/d) bits of working space and O(nd lg lg n σ) time for a selectable parameter d ≥ 1. For the particular case of d = −1 lg n for an arbitrary constant > 0, Kosolobov [25] could improve the running time to O(n(lg σ + lg((lg n)/ ))/ ) for the same space of O( n) bits. Unfortunately, we are unaware of memory-efficient algorithms computing other grammars such as longest-first substitution (LFS) [26], where a modifiable suffix tree is used for computation.…”