Languages with Bounded Multiparty Communication Complexity

Arkadev Chattopadhyay (McGill)

Joint work with:

Andreas Krebs (Tubingen)Michal Koucky (Czech Acad. Sciences)

Mario Szegedy (Rutgers)Pascal Tesson (Laval)Denis Therien (McGill)

‘Number on Forehead’ Model

010001110

Player 1

110111001

Player 2111000001

Player 3

110101111101

Cost of protocol is worst case cost. Dk(f) is the cost of best protocol for f

for the worst partition.

A Theorem for k=2Question. What functions can be computed in

constant communication for the worst partition by two players (denoted by CCC2)?

Remark: A priori there is no reason to believe that CCC2 should have any relationship to space-time complexity classes!

Theorem (Szegedy93). Every function in CCC2 can be computed by linear sized ACC0 circuits.

Three Players

Question: What can we say about CCCk, for k ¸ 3?

We show,Theorem 1. CCC3 contains functions

that have arbitrarily large circuit complexity.

We use a coding trick to show this.

The Coding IdeaLet C : {0,1}* ! {0,1}* be an encoding function.

Definition. For a L, define C(L) as follows: y 2 C(L) if there exists x 2 L s.t. C(x)=y.

Observation. If C has efficient encoding and decoding algorithms, then L and C(L) have comparable complexity.

Fact. If the relative distance of C is more than 2/3, then C(L) 2 CCC3.

Proof of Fact

0100011101001

Player X

110111001

Player Y

1

1

0

111000001

Player Z

Find w² x

Find w ² x

w² y ?

w² z ?

w 2 C(L)?

Such Codes ExistFact: Reed-Solomon codes

concatenated with unary codes can be used to carry out this idea!

Remark: Picking a hard L (that is guaranteed to exist by a counting argument), proves our Theorem.

Question. What makes 3 players so powerful?

¯¯¯¯Pr

x

£X

i

= 1nxi ´ q bjP (x) = a¤

¡ 1=q¯¯¯¯·

°n=cd

Pr[P (x) = a]¯¯¯¯Pr

x

£X

i

= 1nxi ´ q bjP (x) = a¤

¡ 1=q¯¯¯¯·

°n=cd

Pr[P (x) = a]¯¯¯¯Pr

x

£X

i

= 1nxi ´ q bjP (x) = a¤

¡ 1=q¯¯¯¯·

°n=cd

Pr[P (x) = a]

¯¯¯¯Pr

x

£X

i

= 1nxi ´ q bjP (x) = a¤

¡ 1=q

¯¯¯¯·

°n=cd

Pr[P (x) = a]

Key Features of Three Players Every pair of input bits is looked at by some

player.

At least a third of the input bits overlap the view of two players.

Each player knows the precise position in the input word of every input bit that he sees.

Question. How useful is the third feature?

Answer. We obtain insight into this question by considering two simple classes of functions.

Neutral Letter and SymmetricityDefinition. Boolean function f : * ! {0,1} has neutral

letter e if for every x,y 2 *, f(xey) = f(xy).

Theorem 2. Every language with a neutral letter that is in CCCk for some fixed k, is regular. We can also give a decidable algebraic characterization of such languages.

Definition. f over is symmetric if for any permutation and any input string x, f(x)=f((x)).

Theorem 3. A symmetric function f is in CCCk for some fixed k iff it is in CCC2.

Promised Partition Let A be a 0-1 matrix of dimension k£ n.Let there be a promise that each column of A has

at most one 1.Definition. PPartn

k(A) is 1 if each column of A has a 1.

Theorem. PPartnk cannot be computed by k

players using c bits of communication for the row-wise partition of inputs if n ¸ HJ(k,2c).

The above Theorem will give us a handle on languages with a neutral letter in CCCk

Partition to Neutral Letter Let f, g be two functions that have alphabet

= {a,b,e}, where e is neutral.

Let w 2 {a,b}* be the minimal word s.t. f(w) g(w), with |w|=m.

Let D3(f) = D3(g) = c.

Claim: PPartm3 can be computed by three players

using 2c bits of communication for the row-wise partition of input bits.

Proof of the Claim

0100

Player X

1000

Player Y

0001

Player Z

Let w = abab, m=|w|=4

x=ebee

z=eeeb

y=aeee

u = x § y § z = eae bee eee eeb •Compute f(u) =f(abb)•Compute g(u) = g(abb)•Output 1 iff f(u) g(u)

ConsequenceCorollary: If f and g are any two functions over a given

alphabet with a neutral letter, can be computed by k players using c bits of communication and they agree on all inputs of length at most HJ(k, 22c), then they must be identical.

Remark: There are only a finite number of such functions over a given alphabet that can be computed by k players communicating c bits, for each fixed k and c.

Fact: This observation can be used to show that languages in CCCk having a neutral letter, are regular.

Conclusion

We omit the characterization for symmetric functions.

If your are interested, please check out the full version on ECCC at

http://eccc.hpi-web.de/eccc-reports/2006/TR06-117/Paper.pdf

THANK YOU!