lecture24 - COMMUNICATION COMPLEXITY CSci 5403 COMPLEXITY...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XXIV: COMMUNICATION COMPLEXITY x 2 {0,1} n y 2 {0,1} n COMMUNICATION COMPLEXITY ƒ(x,y) ? How many bits do Alice and Bob need to send? Example. ƒ(x,y) = i x i + i y i mod 2 can be computed with two bits: i x i mod 2 i x i + i y i mod 2 Definition. A k-round two party protocol to compute function ƒ: X × Y {0,1} is given by k functions α 0 , α 1 ,…, α k-1 : {0,1}* {0,1}*: x 2 {0,1} n y 2 {0,1} n α 0 (x) α 1 (y, α 0 ) α 2 (x, α 1 , α 0 ) α k-1 (x, α k-2 ,… α 1 , α 0 ) = ƒ(x, y) The transcript, T (x,y), is the messages α 0 ,…, α k-1 . The communication complexity of is C( ) = max x,y {0,1} |T (x,y)| The complexity of ƒ is C(ƒ) = min C( ). Example. ƒ < (x,y): x < y? ƒ (x,y): i x i +y i mod 2 ƒ MAJ (x,y): i (x i +y i ) > n? ƒ DIS (x,y): x y = Ø? ƒ (x,y): i x i y i mod 2 ƒ SAT ( ϕ , ψ ): x,y. ϕ (x) ψ (y) ƒ = (x,y): x y C(ƒ = ) n+1 C(ƒ < ) n+1 C(ƒ ) 2 C(ƒ MAJ ) log n C(ƒ DIS ) n+1 C(ƒ ) n+1 C(ƒ SAT ) 2 Theorem. ƒ. C(ƒ) n+1.
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2 FOOLING SETS Definition. A fooling set for ƒ : X
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