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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

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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:{ 0,1} kn {0,1} 011 001 101100 Alice Bob Chris 101101

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101 00101 01 01 11 10 01 1 0 01 10 101 00101 101 00101 01 01 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 011 001 101100 0101 101 00101 101 101101

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Randomization Randomized Protocols: Each player can use a random source Private Coin / Public Coin 011 001 101100 101101 101 00101 01 01 11 10 01 1 0 01 10

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Why? For k=2 very well understood (“number-on-forehead”=“number-in-hand”). Best known lower bounds: Ω(n/2 k ) [BNS92,CT93,Raz00,FG06] Any function in ACC 0 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,…)

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Natural Questions Does nondeterminism help? Does randomization help? Public coin vs. private coin?

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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)

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Result For k=n O(1) : RP [k] ≠ P [k] * But we don’t know a function that’s in RP [k] - P [k] *

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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].

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Simple Functions Let g:{0,1} n x {0,1} n {0,1} m. 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 7

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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].

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7 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) z z Zzzzz… g(x,y)=z?

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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].

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7 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… 101 00101 01 0 1 11 10 01 1 0 01 101 1 1 1 1 1 1

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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 10100111001 10010101011 011101 0 0 101 00101 01 01 11 10 01 1 0 01 1 01 0 0 7 7

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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].

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# of Protocols vs. # of Functions The Number of Special Protocols: Chris sends a D-bit message that depends on (x,y). Function f C :{0,1} 2n {0,1} D Alice and Bob decide to accept or reject, depending on Chris’ message and (x,z) and (y,z), resp. f A :{0,1} n+m+D {0,1} and f B :{0,1} n+m+D {0,1} log(# functions f C ) = D·2 2n log(# functions f A ) = log(# functions f B ) = 2 n+m+D A protoc. can be described with D·2 2n +2 n+m+D+1 bits.

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# Protocols vs. # of Functions log(#protocols) = D·2 2n +2 n+m+D+1. The Number of Functions: Each simple function is uniquely determined by g:{0,1} n x {0,1} n {0,1} m Each simple function can be described with m·2 2n bits. log(#simple functions) = m·2 2n Putting Things Together: D·2 2n +2 n+m+D+1 ≥ m·2 2n 2 D ≥ m2 2n-n-m-1 -D·2 2n-n-m-1 = (m-D)·2 n-m-1 D ≥ min{m/2, (n-m-2)·log m} E.g., for m=n/2 we have D ≥ n/2

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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].

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Public Coins vs. Private Coins R(f) = complexity for 1-sided error ≤ ½, private coins. R pub (f) = […], public coins. D(f) = complexity of deterministic protocols. Newman ’91: For all functions f: R(f)=R pub (f)+O(log n). Is there a function f, where R(f)=R pub (f )+Ω(log n)? Recall: There is a simple function f* s.t. D ( f*)=Ω(n). Hence, R pub (f*)=O(1) Lemma (similar to k=2): D(f) k(log k)2 O(R(f)) for all f. R(f*) = Ω(log n), if k = n ε, ε<1. R(f*) = R pub (f*)+Ω(log n).

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

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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 2 D messages. Corresponds to a 2 D -coloring of the function matrix of g. x h g 3 3 1 1 2 2 3 3 2 2 0 0 0 0 1 1 2 2 0 0 1 1 3 3 0 0 2 2 3 3 1 1 2 2 3 3 3 3 2 2 1 1 3 3 1 1 0 0 0 0 2 2 2 2 2 2 1 1 3 3 3 3 1 1 0 0 1 1 0 0 2 2 0 0 3 3 1 1 0 0 2 2 2 2 1 1 3 3 3 3 1 1 2 2 3 3 2 2 3 3 2 2 0 0 0 0 1 1 0 0 2 2 3 3 0 0 0 0 1 1 3 3 2 2 3 3 2 2 0 0 0 0 1 1 3 3 3 3 2 2 1 1 0 0 3 3 3 3 2 2 1 1 0 0 1 1 2 2 3 3 1 1 0 0 1 1 1 1 2 2 3 3 0 0 2 2 2 2 3 3 1 1 2 2 3 3 0 0 2 2 1 1 3 3 2 2 1 1 3 3 1 1 2 2 3 3 2 2 0 0 0 0 1 1 2 2 0 0 1 1 3 3 0 0 2 2 3 3 1 1 2 2 3 3 3 3 2 2 1 1 3 3 1 1 0 0 0 0 2 2 2 2 2 2 1 1 3 3 3 3 1 1 0 0 1 1 0 0 2 2 0 0 3 3 1 1 0 0 2 2 2 2 1 1 3 3 3 3 1 1 2 2 3 3 2 2 3 3 2 2 0 0 0 0 1 1 0 0 2 2 3 3 0 0 0 0 1 1 3 3 2 2 3 3 2 2 0 0 0 0 1 1 3 3 3 3 2 2 1 1 0 0 3 3 3 3 2 2 1 1 0 0 1 1 2 2 3 3 1 1 0 0 1 1 1 1 2 2 3 3 0 0 2 2 2 2 3 3 1 1 2 2 3 3 0 0 2 2 1 1 3 3 2 2 1 1

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Proof Idea Consider the most popular value/color pair (z,c). Let M HU 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. x h g 312320012 013023123 321310022 213310102 031022133 123232001 023001323 200133210 332101231 011230223 1230213210 1

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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 11 1 111 0 0 2 1 23202 2302 31023 3202 232311 h g x 31 013023123 32 213310102 03 123232001 023001323 200133210 33 011230223 12 1 11 1 111 0 0 2 1 23202 2302 31023 3202 232311 x xx x xxx 0 0 2 1 23202 2302 31023 3202 2323xx

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Open Problems Define an explicit function in RP [k] – P [k] Prove better lower bounds for simple functions.

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011 001 101100 101101 Alice Bob Chris This talk was not sponsored by Post-It ©

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