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Communication Complexity
In theoretical computer science, communication complexity studies the amount of communication required to solve a problem when the input to the problem is distributed among two or more parties. The study of communication complexity was first introduced by Andrew Yao in 1979, while studying the problem of computation distributed among several machines. The problem is usually stated as follows: two parties (traditionally called Alice and Bob) each receive a (potentially different) n- bit string x and y. The goal is for Alice to compute the value of a certain function, f(x, y), that depends on both x and y, with the least amount of communication between them. While Alice and Bob can always succeed by having Bob send his whole n-bit string to Alice (who then computes the function f), the idea here is to find clever ways of calculating ''f'' with fewer than n bits of communication. Note that, unlike in computational complexity theory, communication complexity is not concerned with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoeffding's Inequality
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963. Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. It is similar to, but incomparable with, one of Bernstein's inequalities. Statement Let be independent random variables such that a_i \leq X_i \leq b_i almost surely. Consider the sum of these random variables, :S_n = X_1 + \cdots + X_n. Then Hoeffding's theorem states that, for all , :\begin \operatorname \left(S_n - \mathrm\left _n \right\geq t \right) &\leq \exp \left(-\frac \right) \\ \operatorname \left(\left , S_n - \mathrm\left _n \right\right , \geq t \right) &\leq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonnegative Rank (linear Algebra)
In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative. For example, the linear rank of a matrix is the smallest number of vectors, such that every column of the matrix can be written as a linear combination of those vectors. For the nonnegative rank, it is required that the vectors must have nonnegative entries, and also that the coefficients in the linear combinations are nonnegative. Formal definition There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative ''m×n''-matrix ''A'' is equal to the smallest number ''q'' such there exists a nonnegative ''m×q''-matrix ''B'' and a nonnegative ''q×n''-matrix ''C'' such that ''A = BC'' (the usual matrix product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two spin states (left-handed and the right-handed circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of multiple states simultaneously, a property that is fundamental to quantum mechanics and quantum computing. Etymology The coining of the term ''qubit'' is attributed to Benjamin Schumacher. In the acknow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collapse Of Randomized Communication Complexity
Collapse or its variants may refer to: Concepts * Collapse (structural) * Collapse (topology), a mathematical concept * Collapsing manifold * Collapse, the action of collapsing or telescoping objects * Collapsing user interface elements ** Accordion (GUI) -- collapsing list items ** Code folding -- collapsing subsections of programs or text ** Outliner -- supporting folding and unfolding subsections * Ecosystem collapse or Ecological collapse * Economic collapse * Gravitational collapse creating astronomical objects * Societal collapse ** Dissolution of the Soviet Union, the collapse of Soviet federalism ** State collapse * Wave function collapse, in physics Medicine and biology In medicine, collapse can refer to various forms of transient loss of consciousness such as syncope, or loss of postural muscle tone without loss of consciousness. It can also refer to: * Circulatory collapse * Lung collapse * Hydrophobic collapse in protein folding Art, entertainment and medi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Fiber
An optical fiber, or optical fibre, is a flexible glass or plastic fiber that can transmit light from one end to the other. Such fibers find wide usage in fiber-optic communications, where they permit transmission over longer distances and at higher Bandwidth (computing), bandwidths (data transfer rates) than electrical cables. Fibers are used instead of metal wires because signals travel along them with less Attenuation, loss and are immune to electromagnetic interference. Fibers are also used for illumination (lighting), illumination and imaging, and are often wrapped in bundles so they may be used to carry light into, or images out of confined spaces, as in the case of a fiberscope. Specially designed fibers are also used for a variety of other applications, such as fiber optic sensors and fiber lasers. Glass optical fibers are typically made by Drawing (manufacturing), drawing, while plastic fibers can be made either by drawing or by extrusion. Optical fibers typically incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can move no faster than the speed of light measured in vacuum. The photon belongs to the class of boson particles. As with other elementary particles, photons are best explained by quantum mechanics and exhibit wave–particle duality, their behavior featuring properties of both waves and particles. The modern photon concept originated during the first two decades of the 20th century with the work of Albert Einstein, who built upon the research of Max Planck. While Planck was trying to explain how matter and electromagnetic radiation could be in thermal equilibrium with one another, he proposed that the energy stored within a material object should be regarded as composed of an integer number of discrete, equal-sized parts. To explain the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Entanglement
Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurement#Quantum mechanics, Measurements of physical properties such as position (vector), position, momentum, Spin (physics), spin, and polarization (waves), polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Problem
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size. The clique problem arises in the following real-world setting. Consider a social network, where the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Entropy (information Theory)
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X, which may be any member x within the set \mathcal and is distributed according to p\colon \mathcal\to[0, 1], the entropy is \Eta(X) := -\sum_ p(x) \log p(x), where \Sigma denotes the sum over the variable's possible values. The choice of base for \log, the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannon (unit), shannons"), while base Euler's number, ''e'' gives "natural units" nat (unit), nat, and base 10 gives units of "dits", "bans", or "Hartley (unit), hartleys". An equivalent definition of entropy is the expected value of the self-information of a v ... [...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] |
