Randomized Algorithms (Spring 2010)/The probabilistic method: Difference between revisions
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== The Basic Idea == | == The Basic Idea == | ||
=== Counting argument === | |||
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|'''Theorem''' | |||
:If <math>{n\choose k}\cdot 2^{1-{k\choose 2}}<1</math> then it is possible to color the edges of <math>K_n</math> vertices with two colors so that there is no monochromatic <math>K_k</math> subgraph. | |||
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|'''Theorem (Shannon)''' | |||
:There is a boolean function <math>f:\{0,1\}^n\rightarrow \{0,1\}</math> with circuit complexity greater than <math>\frac{2^n}{3n}</math>. | |||
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=== Linearity of expectation === | |||
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|'''Theorem''' | |||
:Given an undirected graph <math>G</math> with <math>n</math> vertices and <math>m</math> edges, there is a cut of size at least <math>\frac{m}{2}</math>. | |||
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|'''Theorem''' | |||
:For any set of <math>m</math> clauses, there is a truth assignment that satisfies at least <math>\frac{m}{2}</math> clauses. | |||
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== Alterations == | == Alterations == |
Revision as of 05:34, 16 April 2010
The Basic Idea
Counting argument
Theorem
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Theorem (Shannon)
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Linearity of expectation
Theorem
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Theorem
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