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

Two concrete models Two concrete models for studying complexity: –Decision tree complexity –Communication complexity

Decision Tree Complexity Task: compute f(x) The input x can be accessed by querying x i ’s We only care about the number of queries made Query (decision tree) complexity: min # queries needed. f(x 1,x 2,x 3 )=x 1 ∧ (x 2 ∨ x 3 ) 0 f(x 1,x 2,x 3 )=0 x 2 = ? x 1 = ? 1 0 f(x 1,x 2,x 3 )=1 1 x 3 = ? 01 f(x 1,x 2,x 3 )=0f(x 1,x 2,x 3 )=1

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  x i, z  i: the position we are interested in a: the register holding the queried variable z: other part of the work space –  i,a,z α i,a,z |i, a, z  →  i,a,z α i,a,z |i, a  x i, z  DT D (f), DT R (f), DT Q (f): deterministic, randomized, and quantum query complexities.

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? --- CC D (F) AliceBob F(x,y) xy

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

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…

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

One simple bound Composed functions: F(x,y) = f ∘ g (x,y) = f(g 1 (x (1),y (1) ), …, g n (x (n), y (n) )) –f is an n-bit function, g i is a Boolean function. –x (i) is the i-th block of x. [Thm* 1 ] CC(F) = O(DT(f) max i CC(g i )). –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 z i, she computes g i (x (i),y (i) ) by communicating with Bob. H. Buhrman, R. Cleve, A. Wigderson. STOC, *1. H. Buhrman, R. Cleve, A. Wigderson. STOC, 1998.

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

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 CC D (F) = 1, but DT Q (f) = Ω(n). Similar examples: –f = AND n, g = AND 2, –f = OR n, g = OR 2.

Tightness Question: Can we choose g i ’s s.t. CC(f ∘ g) = Θ(DT(f) max i CC(g i ))? Question: Can we choose g i ’s with O(1) input size s.t. CC(f ∘ g) = Θ(DT(f))? Theorem: Ǝ g i ∊{٧ 2, ٨ 2 } s.t. CC(f ∘ g) = poly ( DT(f)).

More precisely Theorem 1. For all Boolean functions, Theorem 2. For all monotone Boolean functions, –Improve Thm 1 on bounds and range of max. max g i 2 f ^ ; _ g CC R ( f ± g ) = ­ ( DT D ( f ) 1 = 3 ) ; max g i 2 f ^ ; _ g CC Q ( f ± g ) = ­ ( DT D ( f ) 1 = 6 ) : max g 2 f ^ n ; _ n g CC R ( f ± g ) = ­ ( DT D ( f ) 1 = 2 ) ; max g 2 f ^ n ; _ n g CC Q ( f ± g ) = ­ ( DT D ( f ) 1 = 4 ) :

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, max g i 2 f ^ ; _ g CC D ( f ± g ) = O µ max g i 2 f ^ ; _ g CC Q ( f ± g ) 6 ¶ : max g 2 f ^ n ; _ n g CC D ( f ± g ) = O µ max g 2 f ^ n ; _ n g CC Q ( f ± g ) 4 ¶ : 12 Sherstov

Proof [Block sensitivity] –f: function, –x: input, –x I ( I ⊆[n]): flipping variables in I –bs(f,x): max number b of disjoint sets I 1, …, I b flipping each of which changes f-value (i.e. f(x) ≠ f(x I _b )). –bs(f): max x bs(f,x) DT D (f) = O(bs 3 (f)) for general Boolean f, DT D (f) = O(bs 2 (f)) for monotone Boolean f.

Through block sensitivity Goal: Known: DT D (f) = O(bs 3 (f)) for general Boolean f. So it’s enough to prove max g i 2 f ^ ; _ g CC R ( f ± g ) = ­ ( DT D ( f ) 1 = 3 ) ; max g i 2 f ^ ; _ g CC Q ( f ± g ) = ­ ( DT D ( f ) 1 = 6 ) : max g i 2 f ^ ; _ g CC R ( f ± g ) = ­ ( b s ( f )) ; max g i 2 f ^ ; _ g CC Q ( f ± g ) = ­ ( p b s ( f )) :

Disjointness Disj(x,y) = OR(x ٨ y). UDisj(x,y): Disj with promise that |x ٨ y| ≤ 1. Theorem Idea (for our proof): Pick g i ’s s.t. f ∘g embeds an instance of UDisj(x,y) of size bs(f). *1: B. Kalyanasundaram and G. Schintger, SIAMJoDM, Z. Bar-Yossef, T. Jayram, R. Kumar, D. Sivakumar, JCSS, A. Razborov, TCS, *2: A. Razborov, IM, A. Sherstov, SIAMJoC, CC R ( UD i s j ) = £ ( n ) ¤ 1 ; CC Q ( UD i s j ) = £ ( p n ) ¤ 2

bs is Unique OR of flipping blocks Protocol for f(g 1, …, g n ) → Protocol for UDisj b. (b = bs(f)). Input (x’,y’) ∊{0,1} 2n ← Input (x,y) ∊{0,1} 2b –Suppose bs(f) is achieved by z and blocks I 1, …, I b. –i ∉ any block: x’ i = y’ i = z i, g i = ٨. –i ∊ I j : x’ i = x j, y’ i = y i, g i = ٨, if z i = 0 x’ i = ¬x j, y’ i = ¬y i, g i = ٧, if z i = 1 –∃! j s.t. g(x’,y’) = z I _j ⇔ ∃! j s.t. x j ٨ y j = 1. x j ٨ y j = 1 ⇔ g i (x’ i, y’ i ) = ¬z i, ∀ i ∊ I j g i (x’ i, y’ i ) = z i

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 CC Q (f ∘g) = Ω(deg ε (f) ) for some constant size functions g. –Improving the previous: deg ε (f) = Ω(bs(f) 1/2 ) *1: A. Sherstov, SIAMJoC, 2009 *2: T. Lee, S. Zhang, manuscript, 2008.

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

Question: Can we choose g i ’s s.t. CC(f ∘ g) = Θ(DT(f) max i CC(g i ))?