2IND_Luby_Widgerson

2IND_Luby_Widgerson - Pairwise Independence and...

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Pairwise Independence and Derandomization Foundations and Trends in Theoretical Computer Science 2005 Pairwise Independence and Derandomization
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Table of contents Introduction to Pairwise Independence Generating Pairwise Independent Bits Introduction to Derandomizatoin Brute Force Derandomization Derandomization via Pairwise Independence Application- MAXCUT A Complexity Theory Result Pairwise Independence and Derandomization
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Pairwise Independent Events Consider the set of events A 1 , ..., A n The events A 1 , ..., A n are mutually independent if Pr [ A 1 A 2 ... A n ] = Pr [ A 1 ] Pr [ A 2 ] ... Pr [ A n ] The events A i and A j are pairwise independent if Pr [ A i A j ] = Pr [ A i ] Pr [ A j ] for all i , j Pairwise Independence and Derandomization
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Example of Pairwise Independent Events Consider the set Ω = { abc , acb , aaa , bac , bca , bbb , cab , cba , ccc } and probability of any of them being picked is 1 9 . Let event A k denote that the k th letter is a . Clearly Pr [ A k ] = 1 3 Pr [ A 1 A 2 ] = Pr [ A 2 A 3 ] = Pr [ A 3 A 1 ] = 1 9 Pr [ A 1 A 2 A 3 ] = 1 9 6 = Pr [ A 1 ] Pr [ A 2 ] Pr [ A 3 ] Thus, the events pairwise independent but not mutually independent Pairwise Independence is weaker than Mutual Independence Pairwise Independence and Derandomization
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Generating Pairwise Independent Bits We know how to generate mutually independent bits (Coin Flips) Can we use mutually independent bits to generate pairwise independent bits? Construction of
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2IND_Luby_Widgerson - Pairwise Independence and...

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