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On the tightness of Buhrman-Cleve-Wigderson simulation

On the relation between decision tree complexity and communication complexity. On the tightness of Buhrman-Cleve-Wigderson simulation. Shengyu Zhang The Chinese University of Hong Kong. Two concrete models. Two concrete models for studying complexity: Decision tree complexity

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On the tightness of Buhrman-Cleve-Wigderson simulation

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  1. On the relation between decision tree complexity and communication complexity On the tightness of Buhrman-Cleve-Wigderson simulation Shengyu Zhang The Chinese University of Hong Kong

  2. Two concrete models • Two concrete models for studying complexity: • Decision tree complexity • Communication complexity

  3. Decision Tree Complexity • Task: compute f(x) • The input x can be accessed by querying xi’s • We only care about the number of queries made • Query (decision tree) complexity: min # queries needed. f(x1,x2,x3)=x1∧(x2∨x3) x1 = ? 0 1 f(x1,x2,x3)=0 x2 = ? 0 1 x3 = ? f(x1,x2,x3)=1 0 1 f(x1,x2,x3)=0 f(x1,x2,x3)=1

  4. Randomized/Quantum query models • Randomized query model: • We can toss a coin to decide the next query. • Quantum query model: • Instead of coin-tossing, we query for all variables in superposition. • |i, a, z → |i, axi, z • i: the position we are interested in • a: the register holding the queried variable • z: other part of the work space • i,a,zαi,a,z|i, a, z → i,a,zαi,a,z|i, axi, z • DTD(f), DTR(f), DTQ(f): deterministic, randomized, and quantum query complexities.

  5. Communication complexity • [Yao79] Two parties, Alice and Bob, jointly compute a function F(x,y) with x known only to Alice and y only to Bob. • Communication complexity: how many bits are needed to be exchanged? --- CCD(F) x y Alice Bob F(x,y) F(x,y)

  6. Various modes • Randomized: Alice and Bob can toss coins, and a small error probability is allowed. --- CCR(f) • Quantum: Alice and Bob have quantum computers and send quantum messages. --- CCQ(f)

  7. Applications of CC • Though defined in an info theoretical setting, it turned out to provide lower bounds to many computational models. • Data structures, circuit complexity, streaming algorithms, decision tree complexity, VLSI, algorithmic game theory, optimization, pseudo-randomness…

  8. Question: Any relation between the two well-studied complexity measures?

  9. One simple bound • Composed functions: F(x,y) = f∘g (x,y) = f(g1(x(1),y(1)), …, gn(x(n), y(n))) • f is an n-bit function, gi is a Boolean function. • x(i) is the i-th block of x. • [Thm*1] CC(F) = O(DT(f) maxiCC(gi)). • A log factor is needed in the bounded-error randomized and quantum models. • Proof: Alice runs the DT algorithm for f(z). Whenever she wants zi, she computes gi(x(i),y(i)) by communicating with Bob. *1. H. Buhrman, R. Cleve, A. Wigderson. STOC, 1998.

  10. A lower bound method for DT • Composed functions: F(x,y) = f(g1(x(1),y(1)), …, gn(x(n), y(n))) • [Thm] CC(F) = O(DT(f) maxiCC(gi)). • Turning the relation around, we have a lower bound for DT(f) by CC(f(g1, …, gn)): DT(f) = Ω(CC(F)/maxiCC(gi)) • In particular, if |Domain(gi)| = O(1), then DT(f) = Ω(CC(f∘g))

  11. How tight is the bound? • Unfortunately, the bound is also known to be loose in general. • f = Parity, g = ⊕: F = Parity(x⊕y) • Obs: F = Parity(x) ⊕ Parity(y). • So CCD(F) = 1, but DTQ(f) = Ω(n). • Similar examples: • f = ANDn, g = AND2, • f = ORn, g = OR2.

  12. Tightness • Question: Can we choose gi’s s.t. CC(f∘g) = Θ(DT(f) maxiCC(gi))? • Question: Can we choose gi’s with O(1) input size s.t. CC(f∘g) = Θ(DT(f))? • Theorem:Ǝgi∊{٧2,٨2} s.t. CC(f∘g) = poly(DT(f)).

  13. = = R D R D 1 2 1 3 ( ) ( ( ) ) ( ) ( ( ) ) f f C C ­ D T f f C C ­ D T ± m a x g ± m a x g = = ; ; f g f g n n 2 ^ _ 2 ^ _ g g i ; ; = = Q D 1 4 Q D 1 6 ( ) ( ( ) ) ( ) ( ( ) ) f f C C ­ D T f f C C ­ D T ± m a x g ± m a x g = = : : f g f g n n 2 ^ _ 2 ^ _ g g i ; ; More precisely • Theorem 1. For all Boolean functions, • Theorem 2. For all monotone Boolean functions, • Improve Thm 1 on bounds and range of max.

  14. µ µ ¶ ¶ D D Q Q 6 4 ( ( ) ) ( ( ) ) f f f f C C C C O O C C C C ± ± ± ± m m a a x x g g m m a a x x g g = = : : f f g g f f g g n n n n 2 2 2 2 ^ ^ _ _ ^ ^ _ _ g g g g i i ; ; ; ; Implications • A fundamental question: Are classical and quantum communication complexities polynomially related? • Largest gap: quadratic (by Disjointness) • Corollary: For all Boolean functions f, For all monotone Boolean functions f, Sherstov 12

  15. Proof • [Block sensitivity] • f: function, • x: input, • xI (I⊆[n]): flipping variables in I • bs(f,x): max number b of disjoint sets I1, …, Ib flipping each of which changes f-value (i.e. f(x) ≠ f(xI_b)). • bs(f): maxx bs(f,x) • DTD(f) = O(bs3(f)) for general Boolean f, DTD(f) = O(bs2(f)) for monotone Boolean f.

  16. = R R D 1 3 ( ( ) ) ( ( ( ( ) ) ) ) f f b f f C C C C ­ ­ D T ± ± m m a x a x g g s = = ; ; f f g g 2 2 ^ _ ^ _ g g i i ; ; = p Q D 1 6 Q ( ) ( ( ) ) ( ) ( ( ) ) f f C C ­ D T f b f C C ­ ± m a x g ± m a x g s = = : : f g f g 2 ^ _ 2 ^ _ g g i i ; ; Through block sensitivity • Goal: • Known: DTD(f) = O(bs3(f)) for general Boolean f. • So it’s enough to prove

  17. R Q 1 2 p ( ) ( ) ( ) ( ) ¤ ¤ C C U D £ C C U D £ i j i j s n s n = = ; Disjointness • Disj(x,y) = OR(x٨y). • UDisj(x,y): Disj with promise that |x٨y| ≤ 1. • Theorem • Idea (for our proof): Pick gi’s s.t. f∘g embeds an instance of UDisj(x,y) of size bs(f). *1: B. Kalyanasundaram and G. Schintger, SIAMJoDM, 1992. Z. Bar-Yossef, T. Jayram, R. Kumar, D. Sivakumar, JCSS, 2004. A. Razborov, TCS, 1992. *2: A. Razborov, IM, 2003. A. Sherstov, SIAMJoC, 2009.

  18. bs is Unique OR of flipping blocks • Protocol for f(g1, …, gn) →Protocol for UDisjb. (b = bs(f)). • Input (x’,y’)∊{0,1}2n←Input (x,y)∊{0,1}2b • Suppose bs(f) is achieved by z and blocks I1, …, Ib. • i ∉ any block: x’i = y’i = zi, gi = ٨. • i ∊ Ij: x’i = xj, y’i = yi, gi = ٨, if zi = 0 x’i = ¬xj, y’i = ¬yi, gi = ٧, if zi = 1 • ∃! j s.t. g(x’,y’) = zI_j ⇔ ∃! j s.t. xj٨yj = 1. gi(x’i, y’i) = zi xj٨yj = 1 ⇔ gi(x’i, y’i) = ¬zi,∀i∊Ij

  19. Concluding remarks • For monotone functions, observe that each sensitive block contains all 0 or all 1. • Using pattern matrix*1 and its extension*2, one can show that CCQ(f∘g) = Ω(degε(f)) for some constant size functions g. • Improving the previous: degε(f) = Ω(bs(f)1/2) *1: A. Sherstov, SIAMJoC, 2009 *2: T. Lee, S. Zhang, manuscript, 2008.

  20. About the embedding idea • Theorem*1. CCR((NAND-formula ∘ NAND) = Ω(n/8d). • The simple idea of embedding Disj instance was later applied to show depth-independent lower bound: • CCR = Ω(n1/2). • CCQ = Ω(n1/4). • arXiv:0908.4453, with Jain and Klauck. *1: Leonardos and Saks, CCC, 2009. Jayram, Kopparty and Raghavendra, CCC, 2009.

  21. Question: Can we choose gi’s s.t. CC(f∘g) = Θ(DT(f) maxiCC(gi))?

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