l02 - 6.896 Sublinear Time Algorithms February 8, 2007...

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6.896 Sublinear Time Algorithms February 8, 2007 Lecture 2 Lecturer: Ronitt Rubinfeld Scribe: Adriana Lopez In this lecture, we will prove a theorem from last lecture that states that there exists a closeness tester in L 2 . We will also give an algorithm for L 1 -distance testing. Recall the following de±nitions and facts from last lecture: Defnition 1 p x =Pr[ p outputs x ] Defnition 2 (collision probability of p and q ) CP ( p, q )=Pr [ sample from p equals sample from q ]= X x p x q x Defnition 3 (self-collision probability of p ) SCP ( p [ 2 samples from p are equal X x p 2 x = k p k 2 2 1 L 2 -Distance Testing Theorem 4 For all ±, p, q , there exists an O ( ± 4 ) sample tester T such that If k p q k 2 < ± 2 then Pr[ T accepts ] 2 3 . If k p q k 2 then Pr[ T rejects ] 2 3 . We will prove that tester T from last lecture (reproduced below) satis±es the conditions stated in Theorem 4. T ( p, q, ± ): m O ( ± 4 ) S p m samples from p S q m samples from q r p number of self-collisions in S p r q number of self-collisions in S q Q p m (new) samples from p Q q m (new) samples from q r pq number of pairs i, j s.t. i th sample in Q p is equal to j th sample from Q q r 2 m m 1 ( r p + r q ) s 2 r pq iF r s> m 2 ± 2 2 then reject else accept Also recall the following fact from last lecture: E [ r s m 2 k p q k 2 2 Lemma 5 Let b =max x p x ,q x .Th en Var[ r s ] O ( m 3 b 2 + m 2 b ) .
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This note was uploaded on 04/02/2010 for the course CS 6.896 taught by Professor Ronittrubinfeld during the Fall '04 term at MIT.

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l02 - 6.896 Sublinear Time Algorithms February 8, 2007...

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