This paper describes the Lempel-Ziv dimension (Hausdorff like dimension inspired in the LZ78 parsing), its fundamental properties and relation with Hausdorff dimension. It is shown that in the case of individual infinite sequences, the Lempel-Ziv dimension matches with the asymptotical Lempel-Ziv compression ratio. This fact is used to describe results on Lempel-Ziv compression in terms of dime...

We present algorithms computing the non-overlapping Lempel–Ziv-77 factorization and longest previous factor table within small space in linear or near-linear time with help of modern suffix tree representations fitting into limited space. With similar techniques, we show how to answer substring compression queries for Lempel–Ziv-78 a possible logarithmic multiplicative slowdown depending on use...

Combining a note by Rissanen and an idea of enumerative coding we obtain a new implementation of the Ziv-Lempel incremental parsing algorithm for coding and decoding discrete data sequences. Index Terms -Ziv-Lempel algorithm, enumerative coding.

Indexing text collections to support pattern matching queries is a fundamental problem in computer science. New challenges keep arising as databases grow, and for repetitive collections, compressed indexes become relevant. To successfully exploit the regularities of repetitive collections different approaches have been proposed. Some of these are Compressed Suffix Array, Lempel-Ziv, and Grammar...

This paper expands upon existing and introduces new formulations of Bennett's logical depth. In previously published work by Jordon Moser, notions finite-state-depth pushdown-depth were examined compared. These based on finite-state transducers information lossless pushdown compressors respectively. Unfortunately a full separation between the two was not established. formulation restricting how...

Random sequences attain the highest entropy rate. The estimation of entropy rate for an ergodic source can be done using the Lempel Ziv complexity measure yet, the exact entropy rate value is only reached in the infinite limit. We prove that typical random sequences of finite length fall short of the maximum Lempel-Ziv complexity, contrary to common belief. We discuss that, for a finite length,...

A compression algorithm takes a finite structure of a class K as input and produces a finite structure of a different class K’ as output. Given a property P on the class K defined in a logicL, we study the definability of property P on the class K’. We consider two compression schemas on unary ordered structures (words), compression by runlength encoding and the classical Lempel-Ziv. First-orde...