The problem of non-distorting compression (or coding) of sequences of symbols is considered. For sequences of asymptotically zero empirical entropy, a modi0cation of the Lempel–Ziv coding rule is o1ered whose coding cost is at most a 0nite number of times worse than the optimum. A combinatorial proof is o1ered for the well-known redundancy estimate of the Lempel–Ziv coding algorithm for sequenc...

We review several applications of Lempel–Ziv complexity to the characterization of neural responses. In particular, Lempel–Ziv complexity allows to estimate the entropy of binned spike trains in an alternative way to the usual method based on the relative frequencies of words, with the de7nitive advantage of no requiring very long registers. We also use complexity to discriminate neural respons...

It has been suggested that odour encoding in olfactory systems occurs by synchronized firing in neuronal populations. Neurons correlated in terms of the Lempel-Ziv distance of spike trains and the sequential superparamagnetic clustering algorithm belong to the same cluster if they show similar, but not necessarily synchronous, firing patterns. Using multielectrode array recordings from the rat ...

A compression algorithm takes a nite structure of a class K as input and produces a nite structure of a diierent class K' as output. Given a property P on the class K deened in a logic L, we study the deenability of property P on the class K'. We consider two compression schemas on unary ordered structures (words), a naive compression and the classical Lempel-Ziv. First-order properties of stri...

Abstract: If x is a string of nite length over a nite alphabet A, let LZ(x) denote the length of the binary codeword assigned to x by the 1978 version of the Lempel-Ziv data compression algorithm, let t(x) be the number of phrases in the Lempel-Ziv parsing of x, and let (x) be the probability assigned to x by a memoryless source model. Using a very simple technique, we prove the pointwise redun...

This paper describes an eeort to extend the Lempel-Ziv algorithm to a practical universal lossy compression algorithm. It is based on the idea of approximate string matching with a rate-distortion (R ? D) criterion, and is addressed within the framework of vector quantiza-tion (VQ) 4]. A practical one pass algorithm for VQ codebook construction and adaptation for individual signals is developed...

One of the most intricate algorithms related to words is Makanin's algorithm solving word equations. The algorithm is very complicated and the complexity of the problem of solving word equations is not well understood. Word equations can be used to deene various properties of strings, e.g. general versions of pattern-matching with variables. This paper is devoted to introduce a new approach and...

The complexity of computing the Lempel-Ziv factorization and the set of all runs (= maximal repetitions) is studied in the decision tree model of computation over ordered alphabet. It is known that both these problems can be solved by RAM algorithms in O(n log σ) time, where n is the length of the input string and σ is the number of distinct letters in it. We prove an Ω(n log σ) lower bound on ...