Post on 01-Apr-2015
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:
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].
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:
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].
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:
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].
# 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:
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 f of complexity more than n/2-O(1).
f is in co-RP[k] but not in P[k].
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.
011001
101100101101
Alice
Bob
Chris
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