Disperser
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A disperser is a one-sided extractor.[1] 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq \{0,1\}^{m}} we have: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Pr_{U_{m}}[A] > 1 - \epsilon}
Definition (Disperser): A Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (k, \epsilon)} -disperser is a function
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Dis: \{0,1\}^{n}\times \{0,1\}^{d}\rightarrow \{0,1\}^{m}}
such that for every distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0,1\}^{n}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\infty}(X) \geq k} the support of the distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Dis(X,U_{d})} is of size at least Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\epsilon)2^{m}} .
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, e)-extractor is also an (N, M, D, K, e)-disperser.
Other meanings
A disperser is a high-speed mixing device used to disperse or dissolve pigments and other solids into a liquid.
See also
References
- ↑ Shaltiel, Ronen (2002). "Recent developments in explicit constructions of extractors". Bulletin of the EATCS. 77: 67–95. Retrieved 2018-04-10.