Disperser

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 ${\displaystyle A\subseteq \{0,1\}^{m}}$ we have: ${\displaystyle Pr_{U_{m}}[A]>1-\epsilon }$

Definition (Disperser): A ${\displaystyle (k,\epsilon )}$-disperser is a function

${\displaystyle Dis:\{0,1\}^{n}\times \{0,1\}^{d}\rightarrow \{0,1\}^{m}}$

such that for every distribution ${\displaystyle X}$ on ${\displaystyle \{0,1\}^{n}}$ with ${\displaystyle H_{\infty }(X)\geq k}$ the support of the distribution ${\displaystyle Dis(X,U_{d})}$ is of size at least ${\displaystyle (1-\epsilon )2^{m}}$.

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

• Expander graph

References