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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 definitions and facts from last lecture: Definition 1 p x = Pr[ p outputs x ] Definition 2 (collision probability of p and q ) CP ( p, q ) = Pr[ sample from p equals sample from q ] = x p x q x Definition 3 (self-collision probability of p ) SCP ( p ) = Pr[ 2 samples from p are equal ] = x p 2 x = p 2 2 1 L 2 -Distance Testing Theorem 4 For all , p, q , there exists an O ( 4 ) sample tester T such that If p q 2 < 2 then Pr[ T accepts ] 2 3 . If p q 2 > then Pr[ T rejects ] 2 3 . We will prove that tester T from last lecture (reproduced below) satisfies 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 p q 2 2 Lemma 5 Let b = max x p x , q x . Then Var[ r s ] O ( m 3 b 2 + m 2 b ) .
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