l18 - 6.896 Sublinear Time Algorithms November, 16, 2004...

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6.896 Sublinear Time Algorithms November, 16, 2004 Lecture 18 Lecturer: Eli Ben-Sasson Scribe: John Danaher 1 The Blum Luby Rubinfeld (BLR) Linearity Test Throughout our talk today, G, H are groups (not necessarily Abelian), written additively. A function ϕ : G H is called a homomorphism if a, b G, ϕ ( a + b )= ϕ ( a )+ ϕ ( b ). In this lecture we will give a test for closeness to a homomorphism. First we de±ne ”closeness”. Defnition 1 (Distance) The (relative) Hamming distance between two functions f,g : G H ,i s δ ( f,g )=Pr x G [ f ( x ) 6 = g ( x )] . The distance of f from a set of functions F ,is δ ( f,F ) = min f 0 F δ ( f,f 0 ) . Finally, the distance of the set F is δ ( F ) = min f 6 = f 0 F δ ( f,f 0 ) .( I f | F |≤ 1 we de±ne δ ( F )=1 .) We now express our testing problem formally: Defnition 2 (Homomorphism Tester) Given groups G, H , de±ne their set of homomorphisms to be Hom( G, H )= { ϕ : G H : a, b G, ϕ ( a + b )= ϕ ( a )+ ϕ ( b ) } A ( q, ² 0 ,c ) -tester for Hom( G, H ) is a randomized polynomial time algorithm T f with oracle access to a function f : G H , with the following properties. Operation T f tosses some coins, makes q queries to f and outputs accept/reject . Completeness If f Hom( G, H ) then Pr[ T f accepts ]=1 . Soundness Pr[ T f rejects ] > min( c · δ ( f, Hom( G, H )) 0 ) . We would like a tester with the smallest possible query complexity q , and with the largest possible ² 0 ,c (both parameters are at most 1). Our main Theorem, originally proved by Blum, Luby, Rubinfeld [2], shows the existence of a good tester for any pair of groups G, H . The proof of the Theorem we present today is due to Coppersmith. Theorem 3 (BLR Linearity Testing) For every pair of groups G, H , there exists a (3 , 2 / 9 , 1 / 2) - tester for Hom( G, H
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This note was uploaded on 04/02/2010 for the course CS 6.896 taught by Professor Ronittrubinfeld during the Spring '04 term at MIT.

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l18 - 6.896 Sublinear Time Algorithms November, 16, 2004...

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