l04 - 6.896 Sublinear Time Algorithms February 15, 2007...

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6.896 Sublinear Time Algorithms February 15, 2007 Lecture 4 Lecturer: Ronitt Rubinfeld Scribe: Huy Nguyen 1 Testing uniformity of a monotone distribution over a totally ordered set Defnition 1 Let D =1 , 2 , .., n =[ n ] . D is monotone if p 1 p 2 ≥···≥ p n . We want to test whether a monotone distribution over a totally ordered set is uniform or far from uniform. Since all totally ordered sets are equivalent, for simplicity, we assume that the domain is [n]. The key idea of the algorithm is to take samples and check if the number of sample in the left half is close to the number of sample in the right half. Lemma 2 If p ( { 1 ··· k } ) (1 + ± ) p ( { k +1 2 k } ) then | p U [ n ] |≤ ± ProoF δ i ←| p i 1 n | j largest i s.t. p i 1 2 k = 1 n We only consider the case when j k . The case when j>k is similar. We de±ne A 1 ,A 2 3 as follow: A 1 = X i j δ i A 2 = X j<i k δ i A 3 = X n i>k δ i A 1 = A 2 + A 3 because the sum of distribution p and the uniform distribution are both 1.
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l04 - 6.896 Sublinear Time Algorithms February 15, 2007...

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