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Levenshtein Coding
Levenshtein coding is a universal code encoding the non-negative integers developed by Vladimir Levenshtein. Encoding The code of zero is "0"; to code a positive number: #Initialize the step count variable ''C'' to 1. #Write the binary representation of the number without the leading "1" to the beginning of the code. #Let ''M'' be the number of bits written in step 2. #If ''M'' is not 0, increment ''C'', repeat from step 2 with M as the new number. #Write ''C'' "1" bits and a "0" to the beginning of the code. The code begins: To decode a Levenshtein-coded integer: #Count the number of "1" bits until a "0" is encountered. #If the count is zero, the value is zero, otherwise #Start with a variable ''N'', set it to a value of 1 and repeat ''count minus 1'' times: #Read ''N'' bits, prepend "1", assign the resulting value to ''N'' The Levenshtein code of a positive integer is always one bit longer than the Elias omega code of that integer. However, there is a Levenshtein code fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Code (data Compression)
In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic (i.e., ''p''(''i'') ≥ ''p''(''i'' + 1) for all positive ''i''), the expected lengths of the codewords are within a constant factor of the expected lengths that the optimal code for that probability distribution would have assigned. A universal code is ''asymptotically optimal'' if the ratio between actual and optimal expected lengths is bounded by a function of the information entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity. In general, most prefix codes for integers assign longer codewords to larger integers. Such a code can be used to efficiently communicate a message drawn from a set of possible messages, by simply ordering the set of messages b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Levenshtein
Vladimir Iosifovich Levenshtein ( rus, Влади́мир Ио́сифович Левенште́йн, p=vlɐˈdʲimʲɪr ɨˈosʲɪfəvʲɪtɕ lʲɪvʲɪnˈʂtʲejn, a=Ru-Vladimir Iosifovich Levenstein.oga; 20 May 1935 – 6 September 2017) was a Russian scientist who did research in information theory, error-correcting codes, and combinatorial design. Among other contributions, he is known for the Levenshtein distance and a Levenshtein algorithm, which he developed in 1965. He graduated from the ''Department of Mathematics and Mechanics'' of Moscow State University in 1958 and worked at the Keldysh Institute of Applied Mathematics in Moscow ever since. He was a fellow of the IEEE Information Theory Society. He received the IEEE Richard W. Hamming Medal in 2006, for "contributions to the theory of error-correcting codes and information theory, including the Levenshtein distance". Life Levenshtein graduated from Moscow State University in 1958, where he studied in the faculty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0 (number)
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usually by 10. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures. Common names for the number 0 in English are ''zero'', ''nought'', ''naught'' (), ''nil''. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as ''oh'' or ''o'' (). Informal or slang terms for 0 include ''zilch'' and ''zip''. Historically, ''ought'', ''aught'' (), and ''cipher'', have also been used. Etymology The word ''zero'' came into the English language via French from the Italian , a contraction of the Venetian form of Italian via ''ṣafira'' or ''ṣifr''. In pre-Islamic time the word (Arabic ) had the meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Number
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs). Whenever not specifically mentioned, this article adheres to the first convention. In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of the additive inverse (negation, or multiplication by −1) of any object that allows for this construction, and is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate other binary aspects of mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Numeral System
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. Leibniz was specifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Omega Coding
Elias ω coding or Elias omega coding is a universal code encoding the positive integers developed by Peter Elias. Like Elias gamma coding and Elias delta coding, it works by prefixing the positive integer with a representation of its order of magnitude in a universal code. Unlike those other two codes, however, Elias omega recursively encodes that prefix; thus, they are sometimes known as recursive Elias codes. Omega coding is used in applications where the largest encoded value is not known ahead of time, or to compress data in which small values are much more frequent than large values. To encode a positive integer ''N'': #Place a "0" at the end of the code. #If ''N'' = 1, stop; encoding is complete. #Prepend the binary representation of ''N'' to the beginning of the code. This will be at least two bits, the first bit of which is a 1. #Let ''N'' equal the number of bits just prepended, minus one. #Return to Step 2 to prepend the encoding of the new ''N''. To decode an E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Logarithm
In computer science, the iterated logarithm of n, written n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recurrence relation: : \log^* n := \begin 0 & \mbox n \le 1; \\ 1 + \log^*(\log n) & \mbox n > 1 \end On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: :\log^* n = \lceil \mathrm _e(n) \rceil i.e. the base ''b'' iterated logarithm is \log^* n = y if n lies within the interval ^b on the ''x''-axis. In computer science, is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base 2) instead of the natural logarithm (with base ''e''). Mathematically, the iterated logarithm is well-defined for any base greater than e^ \approx 1.444667, not only for base 2 and base ''e''. Analysis of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numeral Systems
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the decimal numeral system (used in common life), the number ''three'' in the binary numeral system (used in computers), and the number ''two'' in the unary numeral system (e.g. used in tallying scores). The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero. Ideally, a numeral system will: *Represent a useful set of numbers (e.g. all integers, or rational numbers) *Give every number represented a unique representation (or at least a standard representation) *Reflect the algebraic and arithme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
