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# l21 - 6.896 Sublinear Time Algorithms Lecture 21 Lecturer...

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6.896 Sublinear Time Algorithms November 30, 2004 Lecture 21 Lecturer: Eli Ben-Sasson Scribe: Mihai Pˇ atra¸ scu 1 Testing Monotonicity Previously, we have seen how to test the linearity of a function. We will now consider another interesting property of a function, namely monotonicity. There have been several results on testing monotonicity; the one we present is from Goldreich et al [1]. In Lecture 1 we have already seen monotonicity testing in one dimension, so now we concentrate on functions with several variables. We will only consider the canonical multidimensional case, the Hamming cube, though this can be generalized. Definition 1 (Distance) The distance between two functions f and g is defined as δ ( f, g ) = Pr x [ f ( x ) = g ( x )] , where x is drawn uniformly from the domain. The distance between f and a set of functions S is δ ( f, S ) = min g S δ ( f, g ) . Definition 2 (Monotonicity) For x, y ∈ { 0 , 1 } n , say x y if ( ) i [ n ] , x i y i . A function f : { 0 , 1 } n → { 0 , 1 } is monotone iff ( ) x, y ∈ { 0 , 1 } n , x y implies f ( x ) f ( y ) . Let M n be the set of monotone boolean functions on n variables. Given n and oracle access to a function f : { 0 , 1 } n → { 0 , 1 } , our problem is to test proximity to M n (in a sense that will be defined below). The tester is quite simple, following the spirit of our linearity test. Monotonicity Tester: Pick x ∈ { 0 , 1 } n and i [ n ] uniformly (and independently) at random. Let y be x with the i -th bit reversed. Accept iff f (min { x, y } ) f (max { x, y } ). Theorem 3 Let ε ( f ) to be the probability that the monotonicity tester rejects f . Then for every Boolean function f : { 0 , 1 } n → { 0 , 1 } , 2 δ ( f, M n ) ε ( f ) δ ( f, M n ) n In this lecture, we will only prove the lower bound on ε ( f ). It can be shown that both inequalities are tight for this algorithm [1]. The lower bound may seem small due to the

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