D. Rubinstein, \Sensitivity vs. block sensitivity of Boolean functions," to appear in Combinatorica. 12  Home/Search   Document Not in Database   Summary   Related Articles   Check

....block sensitivity sensitivity : It is shown in [10] that for all monotone functions, the sensitivity equals the block sensitivity, but for non monotone functions the inequality may be strict. A function with quadratic gap between sensitivity and block sensitivity is exhibited by Rubinstein [14]. Boolean function with block sensitivity b. If a circuit whose gates fail independently with xed probability computes f with error probability at most p, then the number of gates of the circuit is at least b log b) Proof: Let the block sensitivity of f be maximum on input z, and let S 1 ....

D. Rubinstein, \Sensitivity vs. block sensitivity of Boolean functions," to appear in Combinatorica. 12

....sensitivity and shows that it can be exponentially larger than average sensitivity. The natural open question is whether sensitivity and block sensitivity are polynomially related. Gotsman and Linial [5] prove this question equivalent to a seeminglyunrelated problem in graph theory. Rubinstein [10] exhibits a family of functions where the gap is quadratic, and this is the largest known separation. However, the best known upper bound on block sensitivity is exponential in sensitivity this upper bound is a consequence of a result of H. U. Simon [12] relating sensitivity to the number of ....

....clear that s(f) bs 1 (f ) Also, if B is a minimal set such that f(x ) 6= f(x) then f( x ) for all i 2 B. We can conclude that jBj s(f ) and hence that bs(f) bs s(f) f) 2.2. Previous bounds The largest known gap between sensitivity and block sensitivity is due to Rubinstein [10]: Theorem 2.1. There exists a family of functions f for which bs(f) 1 Proof. For any even m, let g(x) be a boolean function on m variables such that g(x) 1 when x 2j Gamma1 = x 2j = 1 for some j and all other input bits are 0. We note that s(g) m, s (g) 1, and bs (g) m=2. ....

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David Rubinstein. Sensitivity vs. block sensitivity of boolean functions. Combinatorica, 15(2):297--299, 1995.

.... ) Clearly, for all x we have s x (f) bs x (f) and bs x (f) C x (f) since a certificate for x will have to contain at least one variable of each sensitive block) Hence: Proposition 1 s(f) bs(f) C(f) The biggest gap known between s(f) and bs(f) is quadratic and was exhibited by Rubinstein [37]: Example 1 Let n = 4k 2 . Divide the n variables in p n disjoint blocks of p n variables: the first block B 1 contains x 1 ; x p n , the second block B 2 contains x p n 1 ; x 2 p n , etc. Define f such that f(x) 1 iff there is at least one block B i where two ....

D. Rubinstein. Sensitivity vs. block sensitivity of Boolean functions. Combinatorica, 15(2):297--299, 1995.

.... bs(f ) Clearly, for all x we have s x (f) bs x (f) and bs x (f) C x (f) since a certificate for x will have to contain at least one variable of each sensitive block) Hence: Proposition 1 s(f) bs(f) C(f) The biggest gap known between s(f) and bs(f) is quadratic, as shown by Rubinstein [Rub95] Example 1 Let n = 4k 2 . Divide the n variables in p n disjoint blocks of p n variables: the first block B 1 contains x 1 ; x p n , the second block B 2 contains x p n 1 ; x 2 p n , etc. Define f such that f(x) 1 iff there is at least one block B i where two ....

D. Rubinstein. Sensitivity vs. block sensitivity of Boolean functions. Combinatorica, 15(2):297--299, 1995.

....sensitivity 3 4 n, and certi cate complexity n 1 (for simplicity, assume that 4 divides n) In the general case the relationships between maximal and block sensitivity are not clear. All we know is that BS(f) and Smax (f) are polynomially related when BS(f) n) 5] Moreover, Rubinstein [8] exhibited an in nite family G of functions with a quadratic gap between the sensitivity and the block sensitivity, and in [3] it was shown that in the average case there is no polynomial (or even exponential) relationship between sensitivity and block sensitivity. In the maximal case it is still ....

D. Rubinstein, \Sensitivity vs. Block Sensitivity of Boolean Functions," in Combinatorica 15, 1995.

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