An Introduction to Randomized algorithms - School of Technology ...
An Introduction to Randomized algorithms - School of Technology ...
An Introduction to Randomized algorithms - School of Technology ...
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Approximating F 2 - algorithm (Alon, Matias, Szegedy)V = {v 1 , . . . , v h }, h = O(n 2 ), each v i - a n-vec<strong>to</strong>r <strong>of</strong> ±1.V is four-wise independent. For v ∈ R V , ∀i 1 ≤ . . . ≤ i 4 ,∀(ɛ 1 , . . . ɛ 4 ) ∈ {−1, 1} 4 , Pr(∀j, v(i j ) = ɛ j ) = 1/16.Choose p ∈ R {1, . . . , h} and s<strong>to</strong>re it using O(log n) bits.v p (i) can be found (for a given i) using only O(log n) bits.Z = ∑ i ɛ im i . Z can be computed in one pass usingO(log n + log m) bits.Compute X = Z 2 . space = O(log m + log n).Take s 1 = 16/λ 2 independent samples X j = X and take theiraverage Y .Take s 2 = 2 log(1/ɛ) independent samples Y i = Y and outputtheir median.
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