Ε-net (computational geometry): Difference between revisions

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{{Other uses|Ε-net (disambiguation){{!}}Ε-net}}

{{lowercase|title=ε-net}}

AnIn [[computational geometry]], an '''''ε''-net''' (pronounced [[epsilon]]-net) in [[computational geometry]] is the approximation of a general set by a collection of simpler subsets. In [[probability theory]] it is the approximation of one probability distribution by another.

An '''''ε''-net''' (pronounced [[epsilon]]-net) in [[computational geometry]] is the approximation of a general set by a collection of simpler subsets. In [[probability theory]] it is the approximation of one probability distribution by another.

== Background ==

[[File:Unit square ɛ-net.svg|right|thumb|An ε-net with ε = 1/4 of the unit square in the range space where the ranges are closed filled rectangles.]]

Let ''X'' be a set and R be a set of subsets of ''X''; such a pair is called a ''range space'' or [[hypergraph]], and the elements of ''R'' are called ''ranges'' or ''hyperedges''. An '''ε-net''' of a subset ''P'' of ''X'' is a subset ''N'' of ''P'' such that any range ''r''&nbsp;∈&nbsp;R with |''r''&nbsp;∩&nbsp;''P''|&nbsp;≥&nbsp;''ε''|''P''| intersects&nbsp;''N''.<ref name="HausslerWelzl1987">{{citation

| last1 = Haussler | first1 = David | author1-link = David Haussler

| last2 = Welzl | first2 = Emo | author2-link = Emo Welzl

| doi = 10.1007/BF02187876

| issue = 2

| journal = [[Discrete and& Computational Geometry]]

| mr = 884223

| pages = 127–151

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For any range space with finite [[VC dimension]] ''d'', regardless of the choice of P, there exists an ε-net of ''P'' of size

: <math>O\left(\frac{d}{\varepsilon} \log \frac{d}{\varepsilon}\right)\!;</math><ref name="HausslerWelzl1987"/>

because the size of this set is independent of ''P'', any set ''P'' can be described using a set of fixed size.

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| doi = 10.1007/BF02570718

| issue = 4

| journal = [[Discrete and& Computational Geometry]]

| mr = 1360948

| pages = 463–479