Separating Deterministic from Randomized Multiparty Communication Complexity Joint work with Paul...

Separating Deterministic from Randomized Multiparty

Communication Complexity

Joint work with

Paul Beame (University of Washington)

Matei David (University of Toronto)

Toni Pitassi (University of Toronto)

Philipp Woelfel

Multiparty Communication

k players

Each player has a Post-It© Note with an n-bit string on the forehead

Each player can see what’s written on the other players’

Post-It© Notes, but not what’s on her own

Goal: compute the function f:0,1kn0,1

011001

101100

Alice

BobChris

101101

101001010101111001100110

10100101101001010101

Protocols

Players communicate in rounds.

In each round one player writes a message to board.

All players can see the messages.

At some point the players agree that the protocol ends.

All players can deduce f(x,y,z) from board contents.

Complexity: Length of the final string on the board

011001

101100

0101

101

00101

101

101101

Randomization

Randomized Protocols:

Each player can use a random source

Private Coin / Public Coin

011001

101100101101

101001010101111001100110

Why?

For k=2 very well understood (“number-on-forehead”=“number-in-hand”).

Best known lower bounds: Ω(n/2k) [BNS92,CT93,Raz00,FG06]

Any function in ACC0 has a protocol with complexity (log n)O(1) for k= (log n)O(1) .

Many other applications (time space tradeoffs, proof system lower bounds, circuit complexity,…)

Natural Questions

Does nondeterminism help?

Does randomization help?

Public coin vs. private coin?

Complexity Classes & Separations

P[k] = class of functions with a k-player deterministic protocol of complexity (log n)O(1)

Analogously define RP[k], BPP[k] , NP[k].

Explicit Separations:

Lee,Shraibman 08 / Chattopadhyay,Ada 08:

Set Inters. NP[k]-BPP[k] for all k ≤ loglog n-O(logloglog n).

David, Pitassi, Viola 08 / Beame,Huynh-Ngoc 08:Explicit functions f NP[k]-BPP[k] for k = Ω(log n)

Result

For k=nO(1): RP[k] ≠ P[k]For k=nO(1): RP[k] ≠ P[k]

*But we don’t know a function that’s in RP[k]-P[k]

*

Proof Overview

Proof for k=3:

1. Define a special class of simple functions.

2. Each simple function is in co-RP[k] .

3. Show: If a simple functions has a deterministic protocol of complexity D, then it has a special deterministic protocol of complexity D + O(1).

4. Show that there are more simple functions than special protocols of complexity n/2.

There exists a simple function of complexity more than n/2-1.

f is in co-RP[k] but not in P[k].

Simple Functions

Let g:0,1n x 0,1n0,1m.

f(x,y,z)=1 if and only if g(x,y)=z.

Chris knows x,y and can compute g(x,y).

E.g., f(x,y,z)=1 iff x+y=z.

29

23

7+23=30.Do I have 30 on my Post-It©?

7+23=30.Do I have 30 on my Post-It©?

7

Proof Overview

Proof for k=3:







7

Each Simple function is in co-RP[k]

Alice knows z. Chris knows g(x,y). Solve: EQ[ g(x,y), z ]. Well-known randomized 2-

party protocol (compare fingerprints).

Small 1-sided error probability, false positives.

Complexity Private coins: O(log n). Public coins: O(1).

29

23

g(x,y)g(x,y)

zz

Zzzzz…Zzzzz…

g(x,y)=z?

Proof Overview

Proof for k=3:







7

Special Protocols for Simple Functions

Let f be a simple fct. and P a det. protocol for f with complexity D

Chris computes r=g(x,y) and writes T=P(x,y,r) on board.

A. & B. check whether they would send the same messages as in T.

If yes, they write 1s, otherwise 0. Iff last 2 bits are 1,

accept. Complexity of

P’ = D+2.30

23

If I have 30 on my Post-It©, then P

produces…

If I have 30 on my Post-It©, then P

produces…

101001010

1011110011001101

1010010101011110011001101

11

11

1 1

Correctness

Case 1: f(x,y,z)=1 g(x,y)=z. Chris sends P(x,y,z). Alice and Bob accept.

Case 2: f(x,y,z)=0 P(x,y,z)≠P(x,y,r) Consider the first bit (at pos. i)

where the protocols differ. This bit is not being sent by Chris’:

Knowing the first i-1 bits of P(x,y,z), Chris cannot distinguish between (x,y,z) and (x,y,r)

Either Alice or Bob notices the error.

27

23

101001110011001010101101110100

1010010101

011110011001101

007

Proof Overview

Proof for k=3:







# of Protocols vs. # of Functions

The Number of Special Protocols:

Chris sends a D-bit message that depends on (x,y).

Function fC:0,12n0,1D

Alice and Bob decide to accept or reject, depending on Chris’ message and (x,z) and (y,z), resp.

fA:0,1n+m+D0,1 and fB:0,1n+m+D0,1

log(# functions fC ) = D·22n

log(# functions fA ) = log(# functions fB ) = 2n+m+D

A protoc. can be described with D·22n+2n+m+D+1 bits.

# Protocols vs. # of Functions

log(#protocols) = D·22n+2n+m+D+1.

The Number of Functions:

Each simple function is uniquely determined by g:0,1n

x 0,1n0,1m

Each simple function can be described with m·22n bits.

log(#simple functions) = m·22n

Putting Things Together:

D·22n+2n+m+D+1 ≥ m·22n

2D ≥ m22n-n-m-1-D·22n-n-m-1 = (m-D)·2n-m-1

D ≥ minm/2, (n-m-2)·log m

E.g., for m=n/2 we have D ≥ n/2

Proof Overview

Proof for k=3:





There exists a simple function f of complexity more than n/2-O(1).


Public Coins vs. Private Coins

R(f) = complexity for 1-sided error ≤ ½, private coins.

Rpub(f) = […], public coins.

D(f) = complexity of deterministic protocols.

Newman ’91: For all functions f: R(f)=Rpub(f)+O(log n).

Is there a function f, where R(f)=Rpub(f)+Ω(log n)?

Recall: There is a simple function f* s.t. D(f*)=Ω(n).

Hence, Rpub(f*)=O(1)

Lemma (similar to k=2): D(f) k(log k)2O(R(f)) for all f.

R(f*) = Ω(log n), if k=nε, ε<1.

R(f*) = Rpub(f*)+Ω(log n).

Explicit Lower Bounds for Simple Functions

Explicit Functions for k=3:

H: 2-wise independent hash family U Z For a hash function hH and key xU, let g(h,x)=h(x).

I.e., f(h,x,z)=1 iff h(x)=z.

Theorem:For k≤εlog n, ε>0 small enough, there is an explicitly defined function fk such that D(fk)=Ω(log n).

Theorem:For k≤εlog n, ε>0 small enough, there is an explicitly defined function fk such that D(fk)=Ω(log n).

Corollary:For k≤εlog n, ε>0 small enough, there is an explicitly defined function fk such that R(fk)=Ω(loglog n) but Rpub(fk)=O(1).

Corollary:For k≤εlog n, ε>0 small enough, there is an explicitly defined function fk such that R(fk)=Ω(loglog n) but Rpub(fk)=O(1).

Proof Idea

Assume there is a protocol with complexity D

Recall: Chris sends message first, then Alice and Bob decide.

For each (h,x) Chris sends one out of 2D messages.

Corresponds to a2D-coloring of the function matrix of g.

xh

g

3 1 2 3 2 0 0 1 2

0 1 3 0 2 3 1 2 3

3 2 1 3 1 0 0 2 2

2 1 3 3 1 0 1 0 2

0 3 1 0 2 2 1 3 3

1 2 3 2 3 2 0 0 1

0 2 3 0 0 1 3 2 3

2 0 0 1 3 3 2 1 0

3 3 2 1 0 1 2 3 1

0 1 1 2 3 0 2 2 3

1 2 3 0 2 1 3 2 1

3 1 2 3 2 0 0 1 2

0 1 3 0 2 3 1 2 3

3 2 1 3 1 0 0 2 2

2 1 3 3 1 0 1 0 2

0 3 1 0 2 2 1 3 3

1 2 3 2 3 2 0 0 1

0 2 3 0 0 1 3 2 3

2 0 0 1 3 3 2 1 0

3 3 2 1 0 1 2 3 1

0 1 1 2 3 0 2 2 3

1 2 3 0 2 1 3 2 1

Proof Idea

Consider the most popular value/color pair (z,c).

Let MHU be the rectangle spanned by these entries.

Assume:

(x,y)M

Chris has entry z

Chris sends message c

Alice and Bob accept.

g(x,y)=z.

xh

g

3 1 2 3 2 0 0 1 2

0 1 3 0 2 3 1 2 3

3 2 1 3 1 0 0 2 2

2 1 3 3 1 0 1 0 2

0 3 1 0 2 2 1 3 3

1 2 3 2 3 2 0 0 1

0 2 3 0 0 1 3 2 3

2 0 0 1 3 3 2 1 0

3 3 2 1 0 1 2 3 1

0 1 1 2 3 0 2 2 3

1 2 3 0 2 1 3 2 101

Proof Idea

Consider function g|M

Hash-Mixing-Lemma [MNT93]:

Pr(g(x,y)=z|(x,y)M) |Z|-1

M is large and only few entries in M have value z.

Same preconditions, but • # of colors reduced by 1.• Some inputs are “covered”

Continue this, until all colors have been used up.

If #colors is too small, not all inputs can be covered.

1

1 1

1

1 1 1

0

0

2

1

2 3 2 0 2

2 3 0 2

3 1 0 2 3

3 2 0 2

2 3 2 3 11

hg x

3 1

0 1 3 0 2 3 1 2 3

3 2

2 1 3 3 1 0 1 0 2

0 3

1 2 3 2 3 2 0 0 1

0 2 3 0 0 1 3 2 3

2 0 0 1 3 3 2 1 0

3 3

0 1 1 2 3 0 2 2 3

1 2

1

1 1

1

1 1 1

0

0

2

1

2 3 2 0 2

2 3 0 2

3 1 0 2 3

3 2 0 2

2 3 2 3 11

x

x x

x

x x x

0

0

2

1

2 3 2 0 2

2 3 0 2

3 1 0 2 3

3 2 0 2

2 3 2 3 xx

Open Problems

Define an explicit function in RP[k] – P[k]

Prove better lower bounds for simple functions.

Separating Deterministic from Randomized Multiparty Communication Complexity Joint work with Paul...

Documents

Transcript of Separating Deterministic from Randomized Multiparty Communication Complexity Joint work with Paul...