Extractor (mathematics)

An 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 (N,M,D,K,\epsilon)} -extractor is a bipartite graph 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 N} nodes on the left and 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 M} nodes on the right such that each node on the left has 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 D} neighbors (on the right), which has the added property that for any subset 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} of the left vertices 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 K} , the distribution on right vertices obtained by choosing a random node in 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} and then following a random edge to get a node x on the right side is 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 \epsilon} -close to the uniform distribution in terms of total variation distance.

File:Extractor (mathematics).svg

For this subset of left vertices (highlighted orange), one edge connected to each vertex is randomly selected (highlighted red), which each correspond to a vertex on the right. This is only one random selection of only one subset; the actual randomness (closeness to the uniform distribution) of a given subset's edge selections can be approximated by running the process many times, and the subset among all possible subsets that yields the least random (furthest from uniform distribution) selection of right vertices is 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 \epsilon} -close to uniform distribution.

A disperser is a related graph.

An equivalent way to view an extractor is as a bivariate 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 E : [N] \times [D] \rightarrow [M]}

in the natural way. With this view it turns out that the extractor property is equivalent to: for any source of randomness 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} 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 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 [T]} .

Extractors are interesting when they can be constructed with small 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,D,\epsilon} relative to 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 N} and 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 M} is as close to 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 KD} (the total randomness in the input sources) as possible.

Extractor functions were originally researched as a way to extract randomness from weakly random sources. See randomness extractor.

Using the probabilistic method it is easy to show that extractor graphs with really good parameters exist. The challenge is to find explicit or polynomial time computable examples of such graphs with good parameters. Algorithms that compute extractor (and disperser) graphs have found many applications in computer science.

References

• Ronen Shaltiel, Recent developments in extractors - a survey