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Las Vegas Algorithm
In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the input. The usual definition of a Las Vegas algorithm includes the restriction that the ''expected'' runtime be finite, where the expectation is carried out over the space of random information, or entropy, used in the algorithm. An alternative definition requires that a Las Vegas algorithm always terminates (is effective), but may output a symbol not part of the solution space to indicate failure in finding a solution. The nature of Las Vegas algorithms makes them suitable in situations where the number of possible solutions is limited, and where verifying the correctness of a candidate solution is relatively easy while finding a solution is complex. Las Vegas algorithms are prominent in the field of artificial intelligence, and in other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, engineering, mathematical, technological and social aspects. Major computing disciplines include computer engineering, computer science, cybersecurity, data science, information systems, information technology and software engineering. The term "computing" is also synonymous with counting and calculating. In earlier times, it was used in reference to the action performed by mechanical computing machines, and before that, to human computers. History The history of computing is longer than the history of computing hardware and includes the history of methods intended for pen and paper (or for chalk and slate) with or without the aid of tables. Computing is intimately tied to the representation of numbers, though mathematical conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Search Problem
In computational complexity theory and computability theory, a search problem is a type of computational problem represented by a binary relation. If ''R'' is a binary relation such that field(''R'') ⊆ Γ+ and ''T'' is a Turing machine, then ''T'' calculates ''R'' if: * If ''x'' is such that there is some ''y'' such that ''R''(''x'', ''y'') then ''T'' accepts ''x'' with output ''z'' such that ''R''(''x'', ''z'') (there may be multiple ''y'', and ''T'' need only find one of them) * If ''x'' is such that there is no ''y'' such that ''R''(''x'', ''y'') then ''T'' rejects ''x'' Intuitively, the problem consists in finding structure "y" in object "x". An algorithm is said to solve the problem if at least one corresponding structure exists, and then one occurrence of this structure is made output; otherwise, the algorithm stops with an appropriate output ("Item not found" or any message of the like). Such problems occur very frequently in graph theory, for example, where searching g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy. The fields of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atlantic City Algorithm
Atlantic City algorithm is a probabilistic polynomial time algorithm that answers correctly at least 75% of the time (or, in some versions, some other value greater than 50%). The term "Atlantic City" was first introduced in 1982 by J. Finn in an unpublished manuscript entitled ''Comparison of probabilistic tests for primality''. Two other common classes of probabilistic algorithms are Monte Carlo algorithms and Las Vegas algorithms. Monte Carlo algorithms are always fast, but only probably correct. On the other hand, Las Vegas algorithms are always correct, but only probably fast. The Atlantic City algorithms, which are bounded probabilistic polynomial time algorithms are probably correct and probably fast. See also * Monte Carlo Algorithm * Las Vegas Algorithm In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microsoft PowerPoint
Microsoft PowerPoint is a presentation program, created by Robert Gaskins and Dennis Austin at a software company named Forethought, Inc. It was released on April 20, 1987, initially for Macintosh computers only. Microsoft acquired PowerPoint for about $14 million three months after it appeared. This was Microsoft's first significant acquisition, and Microsoft set up a new business unit for PowerPoint in Silicon Valley where Forethought had been located. PowerPoint became a component of the Microsoft Office suite, first offered in 1989 for Macintosh and in 1990 for Windows, which bundled several Microsoft apps. Beginning with PowerPoint 4.0 (1994), PowerPoint was integrated into Microsoft Office development, and adopted shared common components and a converged user interface. PowerPoint's market share was very small at first, prior to introducing a version for Microsoft Windows, but grew rapidly with the growth of Windows and of Office. Since the late 1990s, PowerPoint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov's Inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function (mathematics), function of a random variable is greater than or equal to some positive Constant (mathematics), constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in Mathematical analysis, analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Irénée-Jules Bienaymé, Bienaymé's inequality. Markov's inequality (and other similar inequalities) relate probabilities to expected value, expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Statement If is a nonnegative random variable and , then the probability that is at least is at most th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RP (complexity)
In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: * It always runs in polynomial time in the input size * If the correct answer is NO, it always returns NO * If the correct answer is YES, then it returns YES with probability at least 1/2 (otherwise, it returns NO). In other words, the algorithm is allowed to flip a truly random coin while it is running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if the algorithm terminates and produces YES, then the correct answer is definitely YES; however, the algorithm can terminate with NO ''regardless'' of the actual answer. That is, if the algorithm returns NO, it might be wrong. Some authors call this class R, although this name is more commonly used for the class of recursive languages. If the correct answer is YES and the algorithm is run ''n'' times with the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-error Probabilistic Polynomial Time
In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: * It always returns the correct YES or NO answer. * The running time is polynomial in expectation for every input. In other words, if the algorithm is allowed to flip a truly-random coin while it is running, it will always return the correct answer and, for a problem of size ''n'', there is some polynomial ''p''(''n'') such that the average running time will be less than ''p''(''n''), even though it might occasionally be much longer. Such an algorithm is called a Las Vegas algorithm. Alternatively, ZPP can be defined as the class of problems for which a probabilistic Turing machine exists with these properties: * It always runs in polynomial time. * It returns an answer YES, NO or DO NOT KNOW. * The answer is always either DO NOT KNOW or the correct answer. * It returns DO NOT KNOW with probability at mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity classes is a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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