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Lempel–Ziv Complexity
The Lempel–Ziv complexity is a measure that was first presented in the article ''On the Complexity of Finite Sequences'' (IEEE Trans. On IT-22,1 1976), by two Israeli computer scientists, Abraham Lempel and Jacob Ziv. This complexity measure is related to Kolmogorov complexity, but the only function it uses is the recursive copy (i.e., the shallow copy). The underlying mechanism in this complexity measure is the starting point for some algorithms for lossless data compression, like LZ77, LZ78 and LZW. Even though it is based on an elementary principle of words copying, this complexity measure is not too restrictive in the sense that it satisfies the main qualities expected by such a measure: sequences with a certain regularity do not have a too large complexity, and the complexity grows as the sequence grows in length and irregularity. The Lempel–Ziv complexity can be used to measure the repetitiveness of binary sequences and text, like song lyrics or prose. Fractal dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Lempel
Abraham Lempel (; 10 February 1936 – 4 February 2023) was an Israeli computer scientist and one of the fathers of the LZ family of lossless data compression algorithms. Biography Lempel was born on 10 February 1936 in Lwów, Poland (now Lviv, Ukraine). He studied at Technion - Israel Institute of Technology, and received a B.Sc. in 1963, an M.Sc. in 1965, and a D.Sc. in 1967. Since 1977 he held the title of full professor, and was a professor emeritus at Technion. His historically-important works start with the presentation of the LZ77 algorithm in a paper entitled "A Universal Algorithm for Sequential Data Compression" in the ''IEEE Transactions on Information Theory'' (May 1977), co-authored by Jacob Ziv. Lempel was the recipient of the 1998 Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society; and the 2007 IEEE Richard W. Hamming Medal for "pioneering work in data compression, especially the Lempel-Ziv algorithm". Lempel founde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Ziv
Jacob Ziv (; 27 November 1931 – 25 March 2023) was an Israeli electrical engineer and information theorist who developed the LZ family of lossless data compression algorithms alongside Abraham Lempel. He is also a namesake of the Ziv–Zakai bound in estimation theory, with Moshe Zakai. Biography Born in Tiberias, British mandate of Palestine, on 27 November 1931, Ziv received his B.Sc., Dip. Eng. (1954) and M.Sc. degrees (1957) in electrical engineering from the Technion – Israel Institute of Technology, and his D.Sc. degree, receiving the degree from the Massachusetts Institute of Technology in 1962. In 1970, Ziv joined the Technion – Israel Institute of Technology and was the Herman Gross Professor of Electrical Engineering and a Technion Distinguished Professor. Ziv was dean of the Faculty of Electrical Engineering from 1974 to 1976 and vice president for Academic Affairs from 1978 to 1982. From 1987, Ziv had spent three sabbatical leaves at the Information Resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 and is a generalization of classical information theory. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program ''P'' computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than ''P'''s own len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lossless Data Compression
Lossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. Lossless compression is possible because most real-world data exhibits Redundancy (information theory), statistical redundancy. By contrast, lossy compression permits reconstruction only of an approximation of the original data, though usually with greatly improved Bit rate#Bitrates in multimedia, compression rates (and therefore reduced media sizes). By operation of the pigeonhole principle, no lossless compression algorithm can shrink the size of all possible data: Some data will get longer by at least one symbol or bit. Compression algorithms are usually effective for human- and machine-readable documents and cannot shrink the size of random data that contain no Redundancy (information theory), redundancy. Different algorithms exist that are designed either with a specific type of input data in mind or with speci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LZ77 And LZ78
LZ77 and LZ78 are the two lossless data compression algorithms published in papers by Abraham Lempel and Jacob Ziv in 1977 and 1978. They are also known as Lempel-Ziv 1 (LZ1) and Lempel-Ziv 2 (LZ2) respectively. These two algorithms form the basis for many variations including LZW, LZSS, LZMA and others. Besides their academic influence, these algorithms formed the basis of several ubiquitous compression schemes, including GIF and the DEFLATE algorithm used in PNG and ZIP. They are both theoretically dictionary coders. LZ77 maintains a sliding window during compression. This was later shown to be equivalent to the ''explicit dictionary'' constructed by LZ78—however, they are only equivalent when the entire data is intended to be decompressed. Since LZ77 encodes and decodes from a sliding window over previously seen characters, decompression must always start at the beginning of the input. Conceptually, LZ78 decompression could allow random access to the input if the en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lempel–Ziv–Welch
Lempel–Ziv–Welch (LZW) is a universal lossless data compression algorithm created by Abraham Lempel, Jacob Ziv, and Terry Welch. It was published by Welch in 1984 as an improved implementation of the LZ78 algorithm published by Lempel and Ziv in 1978. The algorithm is simple to implement and has the potential for very high throughput in hardware implementations. It is the algorithm of the Unix file compression utility compress and is used in the GIF image format. Algorithm The scenario described by Welch's 1984 paper encodes sequences of 8-bit data as fixed-length 12-bit codes. The codes from 0 to 255 represent 1-character sequences consisting of the corresponding 8-bit character, and the codes 256 through 4095 are created in a dictionary for sequences encountered in the data as it is encoded. At each stage in compression, input bytes are gathered into a sequence until the next character would make a sequence with no code yet in the dictionary. The code for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It is also a measure of the Space-filling curve, space-filling capacity of a pattern and tells how a fractal scales differently, in a fractal (non-integer) dimension. The main idea of "fractured" Hausdorff dimension, dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see #coastline, Fig. 1). In terms of that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, Neuroscience, neurobiology, physics, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a Fair coin, fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a dice, die (which has six equally likely outcomes). Some other important measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
