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 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)

0

f(x1,x2,x3)=0 x2 = ?

x1 = ?

1

0

f(x1,x2,x3)=1

1

x3 = ?

0 1

f(x1,x2,x3)=0 f(x1,x2,x3)=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, axi, 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, axi, z• DTD(f), DTR(f), DTQ(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? --- CCD(F)

Alice Bob

F(x,y) F(x,y)

x y

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)

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(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. H. Buhrman, R. Cleve, A. Wigderson. STOCSTOC, 1998., 1998.

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))

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.

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)).

More precisely

• Theorem 1. For all Boolean functions,

• Theorem 2. For all monotone Boolean functions,

– Improve Thm 1 on bounds and range of max.

maxgi 2 f ;_ g

CCR (f ±g) = (DTD (f )1=3);

maxgi 2 f ;_ g

CCQ (f ±g) = (DTD (f )1=6):

maxg2f ^n ;_ n g

CCR (f ±g) = (DTD (f )1=2);

maxg2 f ^n ;_ n g

CCQ (f ±g) = (DTD (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,

maxgi 2f ;_ g

CCD (f ±g) = Oµ

maxgi 2f ;_ g

CCQ (f ±g)6

¶:

maxg2f n ;_ n g

CCD (f ±g) = Oµ

maxg2f n ;_ n g

CCQ (f ±g)4

¶:12

Sherstov

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.

Through block sensitivity

• Goal:

• Known:

DTD(f) = O(bs3(f)) for general Boolean f.

• So it’s enough to prove

maxgi2f^;_ g

CCR(f ±g)= ­ (DTD (f )1=3);

maxgi2f^;_ g

CCQ(f ±g)= ­ (DTD (f )1=6):

maxgi2f^;_ g

CCR(f ±g)= ­ (bs(f ));

maxgi2f^;_ g

CCQ(f ±g)= ­ (pbs(f )):

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, *1: B. Kalyanasundaram and G. Schintger, SIAMJoDM, 1992. Z. Bar-Yossef, T. Jayram, R. Kumar, D. Sivakumar, JCSS, 2004. A. Razborov, TCS, 1992.T. Jayram, R. Kumar, D. Sivakumar, JCSS, 2004. A. Razborov, TCS, 1992.

*2: A. Razborov, IM, 2003. A. Sherstov, SIAMJoC, 2009.*2: A. Razborov, IM, 2003. A. Sherstov, SIAMJoC, 2009.

CCR(UDisj) = £(n) ¤1; CCQ(UDisj) = £(pn) ¤2

bs is Unique OR of flipping blocks

• Protocol for f(g1, …, gn) → Protocol for UDisjb.(b = bs(f)).

• Input (x’,y’)∊0,12n ← Input (x,y)∊0,12b – 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.

xj٨yj = 1 ⇔ gi(x’i, y’i) = ¬zi, i∀ ∊Ij

gi(x’i, y’i) = zi

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*1: A. Sherstov, SIAMJoC, 2009*2: T. Lee, S. Zhang, manuscript, 2008.*2: T. Lee, S. Zhang, manuscript, 2008.

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: *1: Leonardos and Saks, CCC, 2009. Jayram, Kopparty and Raghavendra, CCC, 2009.

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