Prune And Search
Prune and search is a method of solving optimization problems suggested by Nimrod Megiddo in 1983.Nimrod Megiddo (1983) Linear-time algorithms for linear programming in R3 and related problems. SIAM J. Comput., 12:759–776 The basic idea of the method is a recursive procedure in which at each step the input size is reduced ("pruned") by a constant factor . As such, it is a form of decrease and conquer algorithm, where at each step the decrease is by a constant factor. Let be the input size, be the time complexity of the whole prune-and-search algorithm, and be the time complexity of the pruning step. Then obeys the following recurrence relation: : T(n) = S(n) + T(n(1-p)). This resembles the recurrence for binary search but has a larger term than the constant term of binary search. In prune and search algorithms S(n) is typically at least linear (since the whole input must be processed). With this assumption, the recurrence has the solution . This can be seen either b ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Nimrod Megiddo
Nimrod Megiddo () is a mathematician and computer scientist. He is a research scientist at the IBM Almaden Research Center and Stanford University. His interests include combinatorial optimization, algorithm design and analysis, game theory, and machine learning. He was one of the first people to propose a solution to the bounding sphere and smallest-circle problem. Education Megiddo received his PhD in mathematics from the Hebrew University of Jerusalem for research supervised by Michael Maschler. Career and research In computational geometry, Megiddo is known for his prune and search and parametric search techniques both suggested in 1983Nimrod Megiddo (1983) Linear-time algorithms for linear programming in R3 and related problems. SIAM J. Comput., 12:759–776 and used for various computational geometric optimization problems, in particular to solve the smallest-circle problem in linear time. His former doctoral students include Edith Cohen. Awards and honours Megidd ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Decrease And Conquer Algorithm
A decrease in knitting is a reduction in the number of stitches, usually accomplished by suspending the stitch to be decreased from another existing stitch or by knitting it together with another stitch. Methods of single decreasing (knitting) When more than one stitch is suspended from a stitch, they can hang in different orders. For example, the first stitch could be on top of the second stitch (when seen from the right side) or the reverse, leaning to the left or the right. The order of stitches is important, both for appearance and for the way it pulls the fabric. * K2tog ("knit two together") – Work to the two stitches to be decreased, insert the right-hand needle into the first two stitches as if to knit, wrap yarn around needle in normal manner, slip the two stitches off together and drop them. This creates a right-leaning decrease. * K2tog-L ("knit two together - left") – A left-leaning decrease that is the mirror of K2tog and produces a neater finish than other lef ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Binary Search
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in Time complexity#Logarithmic time, logarithmic time in the Best, worst and average case, worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed fo ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Big Oh Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; one well-known example is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rate ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Master Theorem (analysis Of Algorithms)
Master, master's or masters may refer to: Ranks or titles In education: *Master (college), head of a college *Master's degree, a postgraduate or sometimes undergraduate degree in the specified discipline * Schoolmaster or master, presiding officer of a school In military: * Master (naval), a former naval rank * Master mariner, a licensed mariner who is qualified to be a sea captain in the merchant marine *Master or shipmaster, the sea captain of a merchant vessel * Master-at-arms, a naval police officer, often addressed as "Master" in the Royal Navy In orders and organizations: *Master craftsman, in the Medieval guilds In other: * Master (form of address), an English honorific for boys and young men * Master (judiciary), a judicial official in the courts of common law jurisdictions * Master (Peerage of Scotland), the male heir-apparent or heir-presumptive to a title in the Peerage of Scotland * Master of ceremonies, or MC (emcee), the host of an official public or private stag ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Geometric Series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac14 + \tfrac18 + \cdots is a geometric series with common ratio , which converges to the sum of . Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors. While Ancient Greek philosophy, Greek philosopher Zeno's paradoxes about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematics, Greek mathematicians, for example used by Archimedes to Quadrature of the Parabola, calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance, calculati ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Linear Time
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Minimal Enclosing Sphere
Minimal may refer to: * Minimal (music genre), art music that employs limited or minimal musical materials * "Minimal" (song), 2006 song by Pet Shop Boys * Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland * Minimal (''Dungeons & Dragons''), a creature of magically reduced size in the game ''Dungeons & Dragons'' * Minimal (chocolate), a bean to bar chocolate store in Japan, featured in '' Kantaro: The Sweet Tooth Salaryman'' * Minimal (clothing), an Indonesia clothing-retail company that worked with fashion model Ayu Gani * MINIMAL (restaurant), high end restaurant in Taichung, Taiwan See also * *Minimalism (other) Minimalism is a movement in visual arts, music, and other media that began in post–World War II Western art. Minimalism may also refer to: *Minimalism (computing), a philosophy of programming and configuring computers * Minimalism (philosophy), ... * Maximal (other) * Minimisation (other) * Minimal prime ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |