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Disperser
A disperser is a one-sided extractor. Where an extractor requires that every event gets the same probability under the uniform distribution and the extracted distribution, only the latter is required for a disperser. So for a disperser, an event A \subseteq \^ we have: Pr_ > 1 - \epsilon Definition (Disperser): ''A'' (k, \epsilon)''-disperser is a function'' Dis: \^\times \^\rightarrow \^ ''such that for every distribution'' X ''on'' \^ ''with'' H_(X) \geq k ''the support of the distribution'' Dis(X,U_) ''is of size at least'' (1-\epsilon)2^. Graph theory An (''N'', ''M'', ''D'', ''K'', ''e'')-disperser is a bipartite graph with ''N'' vertices on the left side, each with degree ''D'', and ''M'' vertices on the right side, such that every subset of ''K'' vertices on the left side is connected to more than (1 − ''e'')''M'' vertices on the right. An extractor is a related type of graph that guarantees an even stronger property; every (''N'', ''M'', ''D'', ''K'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extractor (mathematics)
{{Short description, Bipartite graph with nodes An (N,M,D,K,\epsilon) -extractor is a bipartite graph with N nodes on the left and M nodes on the right such that each node on the left has D neighbors (on the right), which has the added property that for any subset A of the left vertices of size at least K, the distribution on right vertices obtained by choosing a random node in A and then following a random edge to get a node x on the right side is \epsilon-close to the uniform distribution in terms of total variation distance. A disperser is a related graph. An equivalent way to view an extractor is as a bivariate function :E : \times \rightarrow /math> in the natural way. With this view it turns out that the extractor property is equivalent to: for any source of randomness X that gives n bits with min-entropy \log K, the distribution E(X,U_D) is \epsilon-close to U_M, where U_T denotes the uniform distribution on /math>. Extractors are interesting when they can be co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness Extractor
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random information entropy, entropy source, together with a short, uniformly random seed, generates a highly random output that appears Independent and identically distributed random variables, independent from the source and Uniform distribution (discrete), uniformly distributed. Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established. For a given source, a randomness extractor can even be considered to be a true random number generator (Hardware_random_number_generator, TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source. Sometimes the term "bias" is used to denote a weakly random sou ... [...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] |
Continuous Uniform Distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra (structure), Boolean algebra under the subset relation, in which the join and meet are given by Intersection (set theory), intersection and Union (set theory), union, and the subset relation itself is the Inclusion (Boolean algebra), Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes. Definitions Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary. Different formalisations of these notions give rise to different notions of expanders: ''edge expanders'', ''vertex expanders'', and ''spectral expanders'', as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible degree. Informal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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