Lempel-Ziv Data Compression Algorithm Research Articles

Predictability measurement for the sequence of travel time by combining the Lempel-Ziv compression and the Longest Common Subsequence algorithm

It is commonly recognized that reliability in travel time is a significant factor to consider, as it not only wields a significant impact on traveller behaviour, but also substantially impacts the general efficiency of transport systems. Predictability measure for Travel time Sequence is probabilistic approach that depends on historical data. To measure the reliability of travel time sequence from the perspective of predictability, entropy plays an important role. To estimate the entropy of such a travel time sequence Lempel-Ziv data compression algorithm can be used but Lempel-Ziv scanned the whole historical string for the exactly identical substring. So, in entropy estimation process some partially similar substring or subsequence gets overlooked. For deep analysis, longest common subsequence (LCS) algorithm is employed to calculate exactly identical substring and most similar subsequence. LCS algorithm discarded redundant subsequence's which further reduce entropy. This lead to increase in predictability of travel time sequence. Thus, the combination of LZ and LCS give more accurate predictability and easily can be implemented for travel time forecasting in ITS to calculate prediction rate for next travel time based on given travel time sequence.

Guessing Individual Sequences: Generating Randomized Guesses Using Finite–State Machines

Motivated by earlier results on universal randomized guessing, we consider an individual–sequence approach to the guessing problem: in this setting, the goal is to guess a secret, individual (deterministic) vector $x^{n}=(x_{1},\ldots,x_{n})$ , by using a finite–state machine that sequentially generates randomized guesses from a stream of purely random bits. We define the finite–state guessing exponent as the asymptotic normalized logarithm of the minimum achievable moment of the number of randomized guesses, generated by any finite–state machine, until $x^{n}$ is guessed successfully. We show that the finite–state guessing exponent of any sequence is intimately related to its finite–state compressibility (due to Lempel and Ziv), and it is asymptotically achieved by the decoder of (a certain modified version of) the 1978 Lempel–Ziv data compression algorithm (a.k.a. the LZ78 algorithm), fed by purely random bits. The results are also extended to the case where the guessing machine has access to a side information sequence, $y^{n}=(y_{1},\ldots,y_{n})$ , which is also an individual sequence.

On Finite Memory Universal Data Compression and Classification of Individual Sequences

Consider the case where consecutive blocks of letters of a semi-infinite individual sequence over a finite-alphabet are being compressed into binary sequences by some one-to-one mapping. No a priori information about is available at the encoder, which must therefore adopt a universal data-compression algorithm. It is known that if the universal Lempel-Ziv (LZ) data compression algorithm is successively applied to -blocks then the best error-free compression, for the particular individual sequence is achieved as tends to infinity. The best possible compression that may be achieved by any universal data compression algorithm for finite -blocks is discussed. It is demonstrated that context tree coding essentially achieves it. Next, consider a device called classifier (or discriminator) that observes an individual training sequence . The classifier's task is to examine individual test sequences of length and decide whether the test -sequence has the same features as those that are captured by the training sequence , or is sufficiently different, according to some appropriate criterion. Here again, it is demonstrated that a particular universal context classifier with a storage-space complexity that is linear in , is essentially optimal. This may contribute a theoretical ldquoindividual sequencerdquo justification for the Probabilistic Suffix Tree (PST) approach in learning theory and in computational biology.

Data compression and genomes: A two-dimensional life domain map

We define the complexity of DNA sequences as the information content per nucleotide, calculated by means of some Lempel–Ziv data compression algorithm. It is possible to use the statistics of the complexity values of the functional regions of different complete genomes to distinguish among genomes of different domains of life (Archaea, Bacteria and Eukarya). We shall focus on the distribution function of the complexity of non-coding regions. We show that the three domains may be plotted in separate regions within the two-dimensional space where the axes are the skewness coefficient and the curtosis coefficient of the aforementioned distribution. Preliminary results on 15 genomes are introduced.

Traveling front solutions to directed diffusion-limited aggregation, digital search trees, and the Lempel-Ziv data compression algorithm.

We use the traveling front approach to derive exact asymptotic results for the statistics of the number of particles in a class of directed diffusion-limited aggregation models on a Cayley tree. We point out that some aspects of these models are closely connected to two different problems in computer science, namely, the digital search tree problem in data structures and the Lempel-Ziv algorithm for data compression. The statistics of the number of particles studied here is related to the statistics of height in digital search trees which, in turn, is related to the statistics of the length of the longest word formed by the Lempel-Ziv algorithm. Implications of our results to these computer science problems are pointed out.

Information-theoretic bounds on average signal transition activity [VLSI systems

Transitions on high-capacitance buses in very large scale integration systems result in considerable system power dissipation. Therefore, various coding schemes have been proposed in the literature to encode the input signal in order to reduce the number of transitions. In this paper, we derive lower and upper bounds on the average signal transition activity via an information-theoretic approach, in which symbols generated by a process (possibly correlated) with entropy rate H are coded with an average of R bits per symbol. The bounds are asymptotically achievable if the process is stationary and ergodic. We also present a coding algorithm based on the Lempel-Ziv data-compression algorithm to achieve the bounds. Bounds are also obtained on the expected number of ones (or zeros). These results are applied to determine the activity-reducing efficiency of different coding algorithms such as, entropy coding, transition signaling, and bus-invert coding, and determine the lower bound on the power-delay product given H and R. Two examples are provided where transition activity within 4% and 9% of the lower bound is achieved when blocks of eight symbols and 13 symbols, respectively, are coded at a time.

More on recurrence and waiting times

Let $\mathbf{X} = {X_n: n = 1, 2,\dots}$ be a discrete valued stationary ergodic process distributed according to probability P. Let $\mathbf{Z}_1^n = {Z_1, Z_2,\dots, Z_n}$ be an independent realization of an n-block drawn with the same probability as X. We consider the waiting time $W_n$ defined as the first time the n-block $\mathbf{Z}_1^n$ appears in X. There are many recent results concerning this waiting time that demonstrate asymptotic properties of this random variable. In this paper, we prove that for all n the random variable $W_nP(Z_1^n)$ is approximately distributed as an exponential random variable with mean 1. We use a Poisson heuristic to provide a very simple intuition for this result, which is then formalized using the Chen-Stein method. We then rederive, with remarkable brevity, most of the known asymptotic results concerning $W_n$ and prove others as well. We further establish the surprising fact that for many sources $W_nP(\mathbf{Z}_1^n)$ is exp(1) even if the probability law for Z is not the same as that of X. We also consider the d-dimensional analog of the waiting time and prove a similar result in that setting. Nearly identical results are then derived for the recurrence time $R_n$ defined as the first time the initial N-block $\mathbf{X}_1^n$ reappears in X. We conclude by developing applications of these results to provide concise solutions to problems that stem from the analysis of the Lempel-Ziv data compression algorithm. We also consider possible applications to DNA sequence analysis.

Improved redundancy of a version of the Lempel-Ziv algorithm

The Lempel-Ziv data compression algorithm has the property that for finite-memory sources the redundancy /spl rho//sub n/ (defined as the difference between the average code rate and the entropy when the memory size is n) is O (log log n/log n). We suggest a new version of the algorithm with redundancy /spl rho//sub n/=O (1/log n).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On Universal Data Compression—An Intuitive Overview

The sliding-window version of the Lempel-Ziv data-compression algorithm (LZ1) has found many applications recently (e.g., the Stacker program for personal computers and the new Microsoft MS-DOS.6.2). Other versions of the Lempel-Ziv data-compression algorithm (LZ2) became an integral part of international standards for data transmission modems and proved themselves to be highly successful. The purpose of this paper is to give an intuitive overview of universal, noiseless data compression of sequences as well as 2-D images, by following the lines of approach which characterizes the family of LZ universal codes and by further extending this approach so as to yield some new results.

The sliding-window Lempel-Ziv algorithm is asymptotically optimal

The sliding-window version of the Lempel-Ziv data-compression algorithm (sometimes called LZ '77) has been thrust into prominence recently. A version of this algorithm is used in the highly successful "Stacker" program for personal computers. If is also incorporated into Microsoft's new MS-DOS-6. Although other versions of the Lempel-Ziv algorithm are known to he optimal in the sense that they compress a data source to its entropy, optimality in this sense has never been demonstrated for this version. In this self-contained paper, we will describe the algorithm, and show that as the "window size," a quantity which is related to the memory and complexity of the procedure, goes to infinity, the compression rate approaches the source entropy. The proof is surprisingly general, applying to all finite-alphabet stationary ergodic sources.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymptotic properties of data compression and suffix trees

Recently, Wyner and Ziv (see ibid., vol.35, p.1250-8, 1989) have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is (1/h) log n in probability where h is the entropy of the alphabet. This finding was used to obtain several insights into certain universal data compression schemes, most notably the Lempel-Ziv data compression algorithm. Wyner and Ziv have also conjectured that their result can be extended to a stronger almost sure convergence. In this paper, we settle this conjecture in the negative in the so called right domain asymptotic, that is, during a dynamic phase of expanding the data base. We prove-under an additional assumption involving mixing conditions-that the length of a typical repeated subword oscillates almost surely (a.s.) between (1/h/sub 1/)log n and (1/h/sub 2/)log n where D<h/sub 2/<h<or=h/sub 1/< infinity . We also show that the length of the nth block in the Lempel-Ziv parsing algorithm reveals a similar behavior. We relate our findings to some problems on digital trees, namely the asymptotic behavior of a (noncompact) suffix tree built from suffixes of a random sequence. We prove that the height and the shortest feasible path in a suffix tree are typically (1/h/sub 2/)log n (a.s.) and (1/h/sub 1/)log n (a.s.) respectively.<<ETX>>

Universal prediction of individual sequences

The problem of predicting the next outcome of an individual binary sequence using finite memory is considered. The finite-state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite-state (FS) predictor. It is proven that this FS predictability can be achieved by universal sequential prediction schemes. An efficient prediction procedure based on the incremental parsing procedure of the Lempel-Ziv data compression algorithm is shown to achieve asymptotically the FS predictability. Some relations between compressibility and predictability are discussed, and the predictability is proposed as an additional measure of the complexity of a sequence. >

Estimating the number of states of a finite-state source

The problem of estimating the number of states of a finite-alphabet, finite-state source is investigated. An estimator is developed that asymptotically attains the minimum probability of understanding the number of states, among all estimators with a prescribed exponential decay rate of overestimation probability. The proposed estimator relies on the Lempel-Ziv data compression algorithm in an intuitively appealing manner. >

New asymptotic bounds and improvements on the Lempel-Ziv data compression algorithm

New results, which follow from a data compression algorithm proposed by A. Lempel and J. Ziv (1977), are presented. The authors describe the original Lempel-Ziv algorithm, and present an improved Lempel-Ziv data compression algorithm. They state and prove some asymptotic bounds on the compression ratio that the improved algorithm achieves. For completeness, they describe the Lempel-Ziv-Welch data compression algorithm presented by T. A. Welch (1984). A software implementation for both the original and the improved Lempel-Ziv algorithms is given, and the algorithms are compared with each other as well as with the Lempel-Ziv-Welch algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fixed data base version of the Lempel-Ziv data compression algorithm

It is demonstrated that a variant of the Lempel-Ziv data compression algorithm where the database is held fixed and is reused to encode successive strings of incoming input symbols is optimal, provided that the source is stationary and satisfies certain conditions (e.g., a finite-order Markov source). A finite memory version of the Lempel-Ziv algorithm compresses (on the average) to about the entropy rate. The necessary memory size depends on the nature of the source.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the estimation of the order of a Markov chain and universal data compression

The authors estimate the order of a finite Markov source based on empirically observed statistics. The performance criterion adopted is to minimize the probability of underestimating the model order while keeping the overestimation probability exponent at a prescribed level. A universal asymptotically optimal test, in the sense just defined, is proposed for the case where a given integer is known to be the upper bound of the true order. For the case where such a bound is unavailable, an alternative rule based on the Lempel-Ziv data compression algorithm is shown to be asymptotically optimal also and computationally more efficient. >