2. 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

3. 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

4. 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

5. 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

6. 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,…)

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

8. 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)

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

10. 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].

11. 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

12. 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].

13. 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?

14. 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].

15. 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

16. 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

17. 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].

18. # 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.

19. # 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

20. 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].

21. 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).

22. Explicit Lower Bounds for Simple Functions Explicit Functions for k=3: H: 2-wise independent hash family U  Z For a hash function hH and key xU, let g(h,x)=h(x). I.e., f(h,x,z)=1 iff h(x)=z. Theorem: For k≤εlog n, ε>0 small enough, there is an explicitly defined function f k such that D ( f k )=Ω(log n). Theorem: For k≤εlog n, ε>0 small enough, there is an explicitly defined function f k such that D ( f k )=Ω(log n). Corollary: For k≤εlog 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).

23. 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

24. Proof Idea Consider the most popular value/color pair (z,c). Let M  HU 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

25. 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

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

27. 011 001 101100 101101 Alice Bob Chris This talk was not sponsored by Post-It ©