her input and the previous messages she has received. A correct protocol for f should terminate for every input pair (x; y) 2 X Y , when both players know f(x; y) and that the protocol has terminated.

The communication complexity of a protocol P is the number of bits exchanged for the worst case input pair. The communication complexity of a Boolean function f : X Y 7! f0; 1g, is that of the best possible protocol for f.

The model was introduced by Yao, 13], and has been studied thoroughly, as has its probabilistic counterpart de ned hereafter. For a survey and exact de nitions see 2], 6]. The communication complexity model has gained increased attention recently due to the generalization of Karchmer and Wigderson, 8], which demonstrates its relation to formula complexity. See also 9].

We focus our attention on probabilistic protocols. In this setting, the

algorithms of the players may be probabilistic; i.e, they may depend on tosses

of random coins and may err. The complexity of a probabilistic protocol

P on input (x; y) , denoted by CP (x; y), is the expected number of bits

exchanged. The refer here to 2],

complexity 6] and 10]

of for

a protocol P is de nitions and

Cdi(sPcu)s=siomn aoxn(txh;ye)CpPro(xb;ayb)i.lisWtiec

communication complexity model.

Two models of access to the random bits are considered in the literature. The PRIVATE, in which each player tosses his private coin, and the COMMON, in which both players share a common random bit string. The PRIVATE model is clearly weaker than the COMMON (since the players may need to communicate their random bits). However, it is more realistic.

Our main objective is to compare the relative power of the two models.

Note that not only the cost but also the output of a probabilistic protocol P on input (x; y), denoted by P(x; y), becomes a random variable. For 0 < 1=2 let

Pcom(f) = fP 2 COMMON j 8(x; y) 2 X Y Prob(P(x; y) 6= f(x; y)) g

Ppri(f) = fP 2 PRIV ATE j 8(x; y) 2 X Y Prob(P(x; y) 6= f(x; y)) g

In words, Pcom(f) and Ppri(f) are the sets of all probabilistic protocols, of the two types, that have error probability of at most . De ne

Ccom(f) = minfC(P ) : P 2 Pcom(f)g; Cpri(f) = minfC(P ) : P 2 Ppri(f)g

2

That is, Ccom(f); (Cpri(f)) is the best complexity that a COMMON, (PRIVATE) protocol can achieve if its error probability is bounded by . Since Ppri(f) Pcom(f) we have Ccom(f) Cpri(f). Our main observa- tion is that the relative power of the two models is nearly the same. This is due to the fact that the model is non-uniform. A similar phenomenon was rst observed by Adleman 1] for the model of Boolean circuits.

Theorem 1.1 Let 0 < 1=2 and 0 < 1 then

C pri (1+

)

(f )

=

O(C com (f )

+

log

n

)

Proof It is enough to uses O(log n ) random

show bits.

that there P can be

is a protocol P directly made to

2woPrk(c1o+mwi)t(hf

) that private

random bits. Player A tosses the necessary random bits, communicates them

to B at a cost of O(log n ) and then they follow P .

The existence of P is asserted by proposition 1.1. 2

Proposition 1.1 Let P 2 Pcom(f); 0 < 1=2 then for any 0 < 1

there only

Ois(laogprno)torcaonldPom2biPts(c.1o+m

)

(f )

such

that

C(P

)

=

O(C (P ))

and

P

uses

Proof Let c = C(P). P may be thought of as a probability distribution on

a nite some

collection of xed random

sdtertienrgm. inFiusrtticheprrmotoorceo,lssinPce=tfhPerregrri==st1

, each Pi de no cost for

ned by random

bits in this model, we may assume that the probability distribution over fPrg

is uniform (by allowing identical copies of protocols in this collection). (We

avoid here questions of irrational values of probability , which are handled in

standard ways in many other discussions on the general theory of probabilistic

algorithms).

Claim 1.1 P1 with the

There is a collection properties:

of

l

=

max( (

n)2 ;

) (n+1)2 c

protocols fPi1; ::; Pilg

=

8(x; y) 2 X Y 1l jfP 2 P1 : P(x; y) 6= f(x; y)gj (1 + ) (1)

8(x; y) 2 X

Y

1 l

P2P1CP (x; y) = O(c)

(2)

3

PoorfrigPoion, failndcPoeiplclkeenctadieocnnotlllyPecat=niodnfwPLritghorrf==et1lq, u=baylmppiarcoxkb(ia(nbng)i2lli;ttyi(.mn+ceW1s)2ea)

protocols out of random protocol will show that L

the out has

properties (1) and (2) with non zero probability. For a xed (x; y) 2 X Y

let A(x; y) P rob(Pr 2

= fPi 2 A(x; y))

Lj

Pi(x; y) 6= f for every

(x; y)g Note that input (x; y). By

by the assumption on P, a Cherno ike inequality

( 11], 4])

P rob(jA(x; y)j (1 + ) l) exp(?2 2 2l) exp(?2n)

Thus, (for n 2 )

Prob(9(x; y); jA(x; y)j (1 + ) l) 2nexp(?2n) < 0:25 (3)

Observe that this corresponds to the probability that L does not have property (1). We need a similar statement for property (2), for which we use Hoe ding inequality

Lemma variables

1.1 Hoe ding(1963), 4], with values in the interval

5] Let Y1; :::; Yl be 0; z]. Let = E(1l

independent random Y jj==l1 i) Then

P

rob(

1 l

Y jj==l1 i

d)

(d?d

(

z? z?d

)z?d

)l=z

For d 2 1; z= ] this simpli es to

P

rob(

1 l

Y jj==l1 i

d)

(

e(d?1) dd

)l

=z

LlCeePmti0(moxua;ryw)L.ithbOezbLs=e=rvnef+Pt1h01;a:t:(;sPYinl10gc(xe. ;

For a xed (x; y) 2 X y); :::; Yl(x; y) meet the n is an upper bound on

Y let Yi(x; y) = assumption of the the complexity of

any protocol for f). Note that E(CPi0(x; y)) = CP (x; y) we have,

= E(1l

jj

=l =1

CP 0 j

(x;

y))

=

1 l

jj

=l =1

E

(CPj0

(x;

y

))

=

CP

(x;

y

)

c

We may assume c < n=3 (otherwise, we are immediately done). We get (for d=3)

8(x; y) 2 X

Y

P

rob(

1 l

jj

=l =1

CP 0 j

(x;

y

)

3c)

(

e2 27

)

nl+c1

(

e2 27

)n+1

3?n?1

4

Thus

Prob(9(x; y);

1 l

jj

=l =1

CP 0 j

(x;

y

)

3c)

2n3?n?1 0:25

(4)

Therefore, there is some choice of L for which both properties (1) and (2)

hold. This completes the proof of the claim. 2

We let P be the probabilistic protocol that picks at random, with uniform distribution, a protocol from the set L. It is guaranteed by claim 1.1 that P has error bound of (1 + ) and it uses only log l = O(log n + log(1=( ))) random bits. This completes the proof of proposition 1.1 2.

Corollary 1.1 Let n?c < < 1=2 ? n?c for some constant c. Then Cpri(f) = O(Ccom(f) + log n)

Proof: Pick so that (1 + ) < 1=2 ? n?c and > n?2c (clearly such

exists). By theorem 1.1, with C(P1) = O(C(P ) +

for log

any P 2 Pcom there is n). The error of P1 can

a protocol P1 2 P be reduced (back)

pri to(1+

),

keeping the same order of magnitude of complexity, by standard ampli cation

methods. ips and

(We repeat P1 a take the majority

constant number of the outputs).

of times 2.

with

independent

coin

protAoncoilms tphoarttaanlwt acylassasnosfweprroctoorcroelcstlyar(eLPas0coVme(gfa)s,).anWdePh0aprvie(f). Those are

Theorem 1.2

C0pri(f) = O(C0com(f) + log n)

Proof: As in the proof of theorem 1.1, except that property (1) holds with probability 1 for every L. So it su ces to pick l = (n + 1)2=c in order to guarantee property (2). 2. Remarks:

1. An important generalization of this classical communication problem is the Karchmer-Wigderson game 8]. Namely, let g : f0; 1gn 7! f0; 1g be a Boolean function, let X = g?1(1) and Y = g?1(0). De ne the relation Rg X Y f1; ::; ng by (x; y; i) 2 Rg i xi 6= yi, where xi; (yi) is

5

the i?th bit of One player get

x (y). input

The communication x 2 X and the other

problem gets y 2

for Y.

Rg is as follows. Their task is to

agree on an i such that (x; is its relation to the formula

y; i) size

2ofRgg.

The importance of this game 8]. Probabilistic protocols are

de ned in a manner similar to the classical case. Our result carries on

to this framework too, by the same proof.

2. In the proofs of theorem 1.1 and theorem 1.2 we only used the inherent power of the nonuniformity of the model, and the fact that the complexity was naturally bounded by n+1. For any nonuniform model with a similar property, our proof asserts that one can use a small amount of randomness, (i.e, the decision tree model, non uniform routing etc.). Indeed the same idea was used in 1], 7].

3. It is well known that for every pair of constants 0 ; 0 if P 2 Pcom(f)

with

C(P

)

=

c,

there

exists

a

P

0

2

P

com 0

(f

)

whose

complexity

is

O(c)

for every (x; y) and all coin tosses. Using this, theorem 1.1 can be

proved just by asserting property (1) (property (2) will hold with prob-

ability 1). However, this does not work for non-Constant error.

4. Our result was recently used in the work of Canetti, and Goldreich, in a study on communication-randomness tradeo s 3].

References 1] L. Adleman, Two theorems om random polynomial time, Proceedings of the 19th Annual IEEE Symposium on Foundation of computer science, 75-83. 2] L. Babai, P. Frankl, J. Simon, Complexity classes in communication complexity theory, Proc. 27th Annual IEEE Symp. on Foundation of computer science, 1986, 337-347 3] R. Canetti, O. Goldreich, Bounds on Tradeo s between randomness and communication complexity, 31th Annual IEEE Symp. on Foundation of computer science, 1990, 767-775.

6

4] W. Hoe ding, Probability inequalities for sums of bounded random variables, American Statistical Association Journal (1963), 13-30 5] M. Hofri, Probabilistic Analysis of algorithms, Springer Verlag, New York, 1987, pp 104 6] L. Lovasz, Communication complexity: a survey, in Path owers and VLSI, (Ed, B. Korte, L.Lovasz, A. Schrijver, ), Springer Verlag, 1990. 7] Krizanc, D. Peleg, E. Upfal, A Time-Randomness Tradeo for oblivious routing, Proc. 20th Annual ACM Symp. on theory of computing, 1988, 93-102. 8] M. Karchmer, A. Wigderson, Monotone circuits for connectivity require super- logarithmic depth, Proceedings of 20th Annual ACM Symposium on Theory of Computing, (1988) 539-550. 9] R. Raz, A. Wigderson, Monotone circuits for matching require linear depth, Proc. 22th Annual ACM Symp. on theory of computing, 1990, 287-292. 10] R. Raz, A. Wigderson, Probabilistic communication complexity of Boolean relations, Proc. 30th Annual IEEE Symp. on Foundation of Computer Science, 1989, 562-567. 11] J. Spencer, Probabilistic methods in combinatorics, to appear in Handbook of combinatorics. 12] A. C. Yao, Probabilistic computation, towards a uni ed measure of complexity, Proc. 18th Annual IEEE Symp. on Foundation of Computer Science, 1977, 222-227. 13] A.C. Yao, Some complexity questions related to distributive computing, Proc. 11th Annual IEEE Symp. on Foundation of computer science, 1979, 151-158 7

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