Post on 05-Feb-2016
description
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))?