separating deterministic from randomized multiparty communication complexity joint work with paul...
TRANSCRIPT
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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