Multiparty Communication Complexity of Disjointness

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1 Multiparty Communication Complexity of Disjointness Aradev Chattopadhyay and Anil Ada School of Computer Science McGill University, Montreal, Canada We obtain a lower bound of Ω Abstract ) n )2 on the -party randomized communication complexity of the Disjointness function in the Number on the Forehead model of multiparty communication. In particular, this yields a bound of n Ω) when is a constant. The previous best lower bound for three players until recently was Ωlog n). Our bound separates the communication complexity classes NP CC and BPP CC for = olog log n). Furthermore, by the results of Beame, Pitassi and Segerlind [4], our bound implies proof size lower bounds for tree-lie, degree threshold systems and superpolynomial size lower bounds for Lovász-Schrijver proofs. Sherstov [6] recently developed a novel technique to obtain lower bounds on two-party communication using the approximate polynomial degree of boolean functions. We obatin our results by extending his technique to the multi-party setting using ideas from Chattopadhyay [8]. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman. Introduction Chandra, Furst and Lipton [7] introduced the Number on the Forehead model of multiparty communication as an extension of Yao s [20] two party communication model. This model, besides being interesting in its own right, has found numerous connections with circuit complexity, proof complexity, branching programs, pseudo-random generators and other areas of theoretical computer science. Both proving upper and lower bounds for this model remain a very challenging tas as it is nown that the overlap of information accessible to players provides significant power to it. In fact, proving a super-polylogarithmic lower bound on the communication needed by poly-logarithmic number of players for computing a function f in the restricted setting of simultaneous deterministic communication, is enough to show that f is not in ACC 0, a class for which no strong bounds are nown. Although several efforts [2, 9, 4, 0] have been made, this goal currently remains out of reach as no superlogarithmic lower bounds exist for even log n players. More modestly, one would lie to be able to determine the communication complexity of simple functions for at least constant number of players. However, despite intensive research see for authors are supported by research grants of Prof. D. Thérien. The first author thans M. David and T. Pitassi for several discussions.

2 example [5, 6, 9, 8]) the best nown lower bounds on the communication complexity of simple functions lie Disjointness and Pointer Jumping was Ωlog n) even for three players. The root cause of this problem is that there was essentially only one method that was the bacbone of almost all strong lower bounds. This method is nown as the discrepancy method and was introduced in the seminal wor of Babai, Nisan and Szegedy [2]. It is however nown that for functions lie Disjointness this method at best yields Ωlog n) lower bounds. Razborov [5] introduced the multi-dimensional discrepancy method to establish a tight relationship between the quantum communication complexity of functions induced by a symmetric base function and the approximation degree of the base function. Recently, Sherstov [6] develops an elegant technique that is simpler and generalizes the results of Razborov by obviating the need for the base function to be symmetric. More importantly for us, the technique in [6] shows that the classical discrepancy method can be modified in a natural way that allows one to obtain strong bounds on two-party quantum communication with bounded error even for functions lie Disjointness that have large discrepancy. In this wor, we suitably modify this technique to extend it to the multi-party setting. In order to achieve this, we use tools developed in Chattopadhyay [8], extending the earlier wor of Sherstov [7], for estimating discrepancy under certain non-uniform distributions. Our result has interesting consequences for communication complexity classes and proof complexity. It provides the first example of an explicit function that has small non-deterministic communication complexity, but exponentially high randomized complexity. In the language of for = olog log n). In fact, the separation is exponential when is any constant. Although such a separation was already nown from the wor of [3], before our wor no explicit function was nown to separate these classes. By the wor of Beame, Pittasi and Szegerlind [4], our lower bounds on the -party complexity of Disjointness implies strong lower bounds on the proof size for a family of proof systems nown as tree-lie, degree threshold systems. Proving lower bounds for these systems was a major open problem in propositional proof complexity. complexity classes, this separates BPP CC. Our Main Result and NP CC Let y,...,y be n-bit binary strings. Define the n boolean matrix A obtained by placing y i in the ith row of A. For x {0, } n, let x y,...,y be the n-bit string x i x i2...x it 0 n t, where i,...,i t are the indices of the all-one columns of A. Let g : {0, } n {, } be a base function. We define G g : {0, }n ) {, } by G g x, y,...,y ) := gx y,...,y ). Observe that G PARITY is the Generalized Inner Product function and G NOR is the Disjointness function. Our main result shows how to use the high approximation degree of a base function to generate a function with high randomized communication complexity. Let R ǫ f) denote the randomized -party communication complexity of f with advantage ǫ. Then, Theorem.. Let f : {0, } m {, } have δ-approximate degree d. Let n 2 2 )e) m d, and f : {0, } n {, } be such that fz) = f z0 n m ). Then R ǫ Gf ) d + logδ + 2ǫ ). 2 2

3 As a corollary we show that R ǫ DISJ ) = Ω n )2 ) for every constant ǫ > 0. In brief, this follows from the following facts. Let NOR n denote the NOR function for inputs of length n. Then f = NOR n and f = NOR m satisfy fz) = f z0 n m ) and by a result of Paturi [3], we now that the /3-approximate degree of NOR m is Θ m). A similar bound for the Disjointness function has been recently and independently obtained by Lee and Shraibman [2]..2 Proof Overview Sherstov [6] devised a novel strategy to mae a passage from approximation degree of boolean functions to lower bounds on two-party communication complexity. We adapt this strategy for our purpose. This adaptation is outlined in Figure. We use three main ingredients, the first of which is the Generalized Discrepancy Method. The classical discrepancy method states that if a function has low discrepancy, then it has high randomized communication complexity. In the generalized discrepancy method this idea is extended as follows: If a function g correlates well with f and has low discrepancy, then f has high randomized communication complexity. The second ingredient is the Approximation/Orthogonality Principle of Sherstov [6]. It states that given a function f with high approximation degree, we can find a function g that correlates well with f, and a distribution µ such that g is orthogonal to every low degree polynomial under µ. The third ingredient, called the Orthogonality-Discrepancy Lemma, is derived from the wor of Chattopadhyay [8]. This taes a function that is orthogonal with low degree polynomials and constructs a new mased function that has low discrepancy. We can then summarize the strategy as follows. We start with a function f : {0, } n {, } with high approximation degree. By the Approximation/Orthogonality Principle, we obtain g that highly correlates with f and is orthogonal with low degree polynomials. From f and g we construct new mased functions F f and F g, similar to the construction of Gf. Since g is orthogonal to low degree polynomials, by the Orthogonality-Discrepancy Lemma we deduce that F g has low discrepancy under an appropriate distribution. Under this distribution F g and F f are highly correlated and therefore applying the Generalized Discrepancy Method, we conclude that F f has high randomized communication complexity. This implies, by the construction of F f, that the randomized communication complexity of G f is high. 2 Preliminaries 2. Multiparty Communication Model In the multiparty communication model introduced by [7], players P,...,P wish to collaborate to compute a function f : {0, } n {, }. The n input bits are partitioned into sets X,...,X [n] and each participant P i nows the values of all the input bits except the ones of X i. This game is often referred to as the Number/Input on the forehead model since it is convenient to picture that player i has the bits of X i written on its forehead, available to everyone 3

4 f high approx-deg Approximation Orthogonality high corr. g E µ gx)px) = 0 for low degp) Orthogonality- Discrepancy F f F g high corr. Generalized Discrepancy Method R ǫf f ) is high R ǫgf ) is high disc F g is low Figure : Proof outline but itself. Players exchange bits, according to an agreed upon protocol, by writing them on a public blacboard. The protocol specifies whose turn it is to spea, and what the player broadcasts as a function of the communication history and the input the player has access to. The protocol s output is a function of what is on the blacboard after the protocol s termination. We denote by D f) the deterministic -party communica tion complexity of f, i.e. the number of bits exchanged in the best deterministic protocol for f on the worst case input. By allowing the players to access a public random string and the protocol to err, one defines the randomized communication complexity of a function. We say that a protocol computes f with ǫ advantage if the probability that P and f agree is at least /2 + ǫ for all inputs. We denote by R ǫ f) the cost of the best protocol that computes f with advantage ǫ. One further introduces non-determinism in protocols by allowing God to help the players by furnishing a proof string. As is usual with non-determinism in other models, a correct non-deterministic protocol P for f has the following property: on every input x at which fx) =, Px, y) = for some proof string y and whenever fx) =, Px, y) = for all proof strings y. The length of the proof string y is now included in the cost of P on an input and N f) denotes the cost of the best non-deterministic protocol for f on the worst input. Communication complexity classes were introduced for two players in [] in which efficient protocol was defined to have cost no more than polylogn). This idea naturally extends to the multiparty model giving rise to the following classes: P CC := {f D f) = polylogn)}, BPP CC := {f R /3 f) = polylogn)} and NP CC := {f N f) = polylogn)}. Determining the relationship among these classes is an interesting research theme within the broader area of understanding the relative power of determinism, non-determinism and randomness in computation. While Beame et.al. [3] show that BPP CC NP CC, no explicit function was nown that separated these classes. 2.2 Cylinder Intersections and Discrepancy The ey combinatorial object that arises in the study of multiparty communication is a cylinderintersection. A -cylinder in the ith dimension is a subset S of Y Y with the property that membership in S is independent of the ith coordinate. A set S is called a cylinder-intersection if 4

5 S = i= S i, where S i is a cylinder in the ith dimension. One can represent a -cylinder in the ith dimension by its characteristic function φ i : {0, } n ) {0, }. Here φ i y,..., y ) does not depend on y i. A cylinder intersection is represented as the product φy,..., y ) = φ y,..., y )...φ y,..., y ). It is well nown that a protocol that computes f with cost c partitions the input space of f into at most 2 c monochromatic cylinder intersections. An important measure, defined for a function f : Y... Y {, }, is its discrepancy. With respect to any probability distribution µ over Y Y and cylinder intersection φ, define disc φ,µ f) = [ Pr fy,...,y ) = φy,...,y ) = ] µ [ Pr fy,...,y ) = φy,...,y ) = ] µ. Since f is -/ valued, it is not hard to verify that equivalently: disc φ,µ f) = E y,...,y µfy,...,y )φy,...,y ). ) The discrepancy of f w.r.t. µ, denoted by disc,µ f) is max φ disc φ,µ f). For removing notational clutter, we often drop µ from the subscript when the distribution is clear from the context. We now state the discrepancy method which connects the discrepancy and the randomized communication complexity of a function. Theorem 2. see [2, ]). Let 0 < ǫ /2 be any real and 2 be any integer. For every function f : Y... Y {, } and distribution µ on inputs from Y Y, ) R ǫ f) log 2ǫ. 2) disc,µ f) 2.3 Fourier Expansion We consider the vector space of functions from {0, } n R. Equip this space with the standard inner product f, g f, g = E x U fx)gx) 3) For each S [n], define χ S x) = ) P i S x i. Then it is easy to verify that the set of functions {χ S S [n]} forms an orthonormal basis for this inner product space, and so every f can be expanded in terms of its Fourier coefficients fx) = S [n] ˆfS)χ S x) 4) where ˆfS) is defined as f, χ S. This expansion is unique and the exact degree of f is defined to be the largest d such that there exists S [n] with S = d and ˆfS) 0. 5

6 2.4 Approximation Degree A natural question is the following. How large degree is needed if we want to simply approximate f well? Define the ǫ-approximate degree of f, denoted by deg ǫ f) to be the smallest integer d for which there exists a function φ of exact degree d such that max x {0,} n For any D : {0,,...,n} {, }, define fx) φx) ǫ l 0 D) {0,,..., n/2 } l D) {0,,..., n/2 } such that D is constant over the interval [l 0 D), n l D)] and l 0 D) and l D) are the smallest possible values for which this happens. Paturi s theorem provides bounds on the approximate degree of symmetric functions. Theorem 2.2 Paturi[3]). Let f : {0, } n {, } be any symmetric function induced from the predicate D : {0,...,n} {, }. Then, deg /3 f) = Θ nl0 D) + l D)) ) 5) In particular, the /3-approximate degree of NOR is Θ n). 3 The Generalized Discrepancy Method Babai, Nisan and Szegedy [2] estimated the discrepancy of functions lie GIP w.r.t -wise cylinder intersections and the uniform distribution. These estimates resulted in the first strong lower bounds in the -party model via Theorem 2.. Unfortunately, the applicability of Theorem 2. is limited to those functions that have small discrepancy. Disjointess is a classical example of a function that does not have small discrepancy. Lemma 3. Follore). Under every distribution µ over the inputs, disc,µ DISJ ) = Ω/n). Proof. Let X + and X be the set of disjoint and non-disjoint inputs respectively. The first thing to observe is that if µx + ) µx ) = Ω/n), then we are done immediately by considering the discrepancy over the intersection corresponding to the entire set of inputs. Hence, we may assume µx + ) µx ) = o/n). Thus, µx ) /2 o/n). However, X can be covered by the following n monochromatic cylinder intersections: let C i be the set of inputs in which the ith column is an all-one column. Then X = n i= C i. By averaging, there exists an i such that µc i ) /2n o/n 2 ). Taing the discrepancy of this C i, we are done. It is therefore impossible to obtain better than Ωlog n) bounds on the communication complexity of Disjointness by a direct application of the discrepancy method. In fact, the above argument shows that Theorem 2. fails to give better than polylogarithmic lower bound for every function that is in NP CC or co-np CC. 6

7 Sherstov [6, Sec 2.4] provides a nice reinterpretation of Razborov s discrepancy method for two party quantum communication complexity by pointing out the following: in order to prove a lower bound on the communication complexity of a function f in any bounded error model, it is sufficient to find a function g that correlates well with f under some distribution but has large communication complexity. Based on this observation, we modify the discrepancy method to the following: Lemma 3.2 Generalized Discrepancy Method). Denote X = Y... Y. Let f : X {, } and g : X {, } be such that under some distribution µ we have Corr µ f, g) δ. Then ) δ + 2ǫ R ǫ f) log 6) disc,µ g) Proof. Let P be a -party randomized protocol that computes f with advantage ǫ and cost c. Then for every distribution µ over the inputs, we can derive a deterministic -player protocol P for f that errs only on at most /2 ǫ fraction of the inputs w.r.t. µ) and has cost c. Tae µ to be a distribution satisfying the correlation inequality. We now P partitions the input space into at most 2 c monochromatic w.r.t. P ) cylinder intersections. Let C denote this set of cylinder intersections. Then, δ E x µ fx)gx) = x fx)gx)µx) x P x)gx)µx) + x fx) P x))gx)µx) Since P is a constant over every cylinder intersection S in C, we have δ S C P x)gx)µx) + x S x gx)µx) + S C x S x 2 c disc,µ g) + 2/2 ǫ). gx) fx) P x) µx) fx) P x) µx) This gives us immediately 6). Observe that when f = g, i.e. Corr µ f, g) =, we get the classical discrepancy method Theorem 2.). 4 Generating Functions With Low Discrepancy 4. Masing Schemes We have already defined one masing scheme through the notation x y,...,y. This allowed us to define G g for a base function g. Well-nown functions such as GIP and DISJ are respresentable in this notation by G PARITY and G NOR respectively. We now define a second masing scheme which plays a crucial role in lowerbounding the communication complexity of G g. This masing scheme is obtained by first slightly simplifying the pattern matrices in [6] and then generalizing the simplified matrices to higher dimension for dealing with multiple players. 7

8 S x = S x S, S 2 = 00 Figure 2: Illustration of the masing scheme x S, S 2. The parameters are l = 3, m = 3, n = 27. Let S,...S [l] m for some positive l and m. Let x {0, } n where n = l m. Here it is convenient to thin of x to be divided into m equal blocs where each bloc is a -dimensional array with each dimension having size l. Each S i is a vector of length m with each co-ordinate being an element from {,...,l}. The vectors S,...,S jointly unmas m bits of x, denoted by x S,...,S, precisely one from each bloc of x i.e. x[][s [], S 2 [],..., S []],...,x[m][s [m], S 2 [m],...,s [m]]. where x[i] refers to the ith bloc of x. See Figure 2 for an illustration of this masing scheme. For a given base function f : {0, } m {, }, we define F f : {0, }n [l] m ) {, } as F f x, S,...,S ) = fx S,...,S ). Lemma 4.. If f : {0, } m {, } and f : {0, } n {, } have the property that fz) = f z0 n m ) here n = l m as described in the construction of F f ), then R ǫ F f ) Rǫ Gf ). 7) Proof Setch. Observe that there are functions Γ i : [l] m {0, } n such that F f x, S,...,S ) = G f x,γ S ),...,Γ S )) for all x, S,...,S. Therefore the players can privately convert their inputs and apply the protocol for G f. Note that the proof shows 7) holds not just for randomized but any model of communication. 4.2 Orthogonality and Discrepancy Now we prove that if the base function f in our masing scheme has a certain nice property, then the mased function F f has small discrepancy. To describe the nice property, let us define the following: for a distribution µ on the inputs, f is µ, d)-orthogonal if E x µ fx)χ S x) = 0, for all S < d. Then, Lemma 4.2 Orthogonality-Discrepancy Lemma). Let f : {, } m {, } be any µ, d)- orthogonal function for some distribution µ on {, } m and some integer d > 0. Derive the probability distribution λ on {, } n [l] m) from µ as follows: λx, S,...,S ) = µx S,...,S ). l m ) 2 n m Then, disc,λ F f ) ) 2 )m j=d ) ) )m 2 2 j 8) j l 8

9 Hence, for l 22 )em d and d > 2, disc,λ F f ). 9) d/2 2 Remar. The Lemma above appears very similar to the Multiparty Degree-Discrepancy Lemma in [8] that is an extension of the two party Degree-Discrepancy Theorem of [7]. There, the magic property on the base function is high voting degree. It is worth noting that µ, d)-orthogonality of f is equivalent to voting degree of f being at least d. Indeed the proof of the above Lemma is almost identical to the proof of the Degree-Discrepancy Lemma save for the minor details of the difference between our masing scheme and the one used in [8]. Proof of Lemma 4.2. The starting point is to write the expression for discrepancy w.r.t. an arbitrary cylinder intersection φ, disc φ F f ) = F f x, S,...,S )φx, S,...,S ) λx, S,...,S ) 0) x,s,...,s This changes to the more convenient expected value notation as follows: disc φ F f ) = 2m E x,s,...,s F f x, S,...,S ) φx, S,...,S )µ x S,...,S ) ) where, x, S,...,S ) is now uniformly distributed over {0, } l m [l] m). Then, we use the tric of repeatedly combining triangle inequality with Cauchy-Schwarz exactly as done in Chattopadhyay[8] or even before by Raz[4]) to obtain the following: disc φ F f ))2 2 2 m E S 0,S,...,S 0,S H f S 0, S,...,S0, S ) 2) where, H f S 0, S,...,S0, S ) = E x {0,} l m F f x, S u,...,su )µx Su,...,Su )) 3) u {0,} We loo at a fixed S0 i, Si, for i =,...,. Let r i = S 0 i Si and r = i r i for i 2. We now mae two claims that are analogous to Claim 5 and Claim 6 respectively in [8]. Claim 4.3. Claim 4.4. Let r < d. Then, H f S 0, S,...,S0, S ) 2 2 )r 2 2 m 4) H f S 0, S,...,S0, S ) = 0 5) 9

10 We prove these claims in the next section. Claim 4.3 simply follows from the fact that µ is a probability distribution and f is /- valued while Claim 4.4 uses the µ, d)-orthogonality of f. We now continue with the proof of the Orthogonality-Discrepancy Lemma assuming these claims. Applying them, we obtain disc φ F f ))2 )m j=d 2 2 )j j + +j =j Pr [ r = j r = j ] 6) Substituting the value of the probability, we further obtain: disc φ F f ))2 )m j=d 2 2 )j j + +j =j m j ) m j The following simple combinatorial identity is well nown: ) ) ) m m )m = j j + +j =j j j ) l ) m j l ) m j l )m 7) Plugging this identity into 7) immediately yields 8) of the Orthogonality-Discrepancy Lemma. Recalling ) )m j e )m ) j, j and choosing l 2 2 )em/d, we get 9). 4.3 Proofs of Claims We identify the set of all assignments to boolean variables in X = {x,...,x n } with the n-ary boolean cube {0, } n. For any u {0, }, let Z u represent the set of m variables indexed jointly by Su,...,Su. There is precisely one variable chosen from each bloc of X. Denote by Z i [α] the unique variable in Z i that is in the αth bloc of X, for each α m. Let Z = u Z u. We abuse notation for the sae of clarity and use Z u in the context of expected value calculations to also mean a uniformly chosen random assignment to the variables in the set Z u. Proof of Claim 4.4. S 0, S,...,S0, S ) = E Z fz 0 0 )µz 0) E X Z0 H f u {0,} u 0 fz u )µz u ) 8) Observe that for any bloc α and any u 0, Z u [α] = Z 0 [α] iff for each i such that u i =, S0 i[α] = Si [α]. Recall that r i is the number of indices α such that S0 i[α] = Si [α]. Therefore, there are at most r = i= r i many indices α such that Z u [α] = Z 0 [α] for some u 0. This means the inner expectation in 8) is a function that depends on at most r variables. Since f is orthogonal under µ with every polynomial of degree less than d and r < d, we get the desired result. 0

11 Proof of Claim 4.3. Observe that since F f is /- valued, we get the following: H f S 0, S,...,S0, S ) Ex µx Su,...,S u {0,} = E X Z E Z u {0,} µz u ) = E X Z 2 Z Z {0,} Z E X Z 2 Z y,...,y {0,} m u ) u {0,} µz u ) 9) i= µy i ) 20) where the last inequality holds because every product in the inner sum of 9) appears in the inner sum of 20). Using the fact that µ is a probability distribution, we get: RHS of 20) = E X Z 2 Z = E X Z 2 Z = 2 Z. i= y i {0,} m µy i ) We now find a lower bound on Z. Let t u denote the Hamming weight of the string u and {j,...,j tu } denote the set of indices in [ ] at which u has a. Define Y u = { Z u [α] S js [α] Sjs 0 [α]; s t i; α m } 2) The following follow from the above definition. Y 0 = m and Y u m s t i r js m r for all u 0. Y u Y v =, for u v. This follows from the following argument: wlog assume there is an index β where u has a one but v has a zero. Consider any bloc α such that Z u [α] is in Y u. It must be true that S β [α] Sβ 0 [α]. This means that Z u[α] Z v [α]. Therefore Z u [α] is not in Y v and we are done. Y := u {0,} Y u = Z. This is because if Z u [α] is not in Y u then there are indices j,...,j s where u contains a one and S j i 0 [α] = Sj i [α]. Let v be the string that contains a zero at positions j,...,j s and at other positions, corresponds to u. Then by definition, Z u [α] = Z v [α] Y v. Thus, Z = Y = u Y u m+ u 0 m r) = 2 m 2 )r and the result follows. 5 The Main Result Before proving the main result, we borrow from Sherstov [6] a beautiful duality between approximability and orthogonality. The intuition is that if a function is at a large distance from the linear space spanned by the characters of degree less than d, then its projection on the dual space spanned by characters of degree at least d is large. More precisely,

12 Lemma 5.. Let f : {, } m R be given with deg δ f) = d. Then there exists g : {, } m {, } and a distribution µ on {, } m such that g is µ, d)-orthogonal and Corr µ f, g) > δ. We do not prove this Lemma but the interested reader can read its short proof in [6] which is based on an application of linear programming duality. Theorem 5.2 Main Theorem). Let f : {0, } m {, } have δ-approximate degree d. Let n 2 2 )e) m d, and f : {0, } n {, } be such that fz) = f z0 n m ). Then R ǫ Gf ) d + logδ + 2ǫ ). 22) 2 Proof. Applying Lemma 5. we obtain a function g and a distribution µ such that Corr µ f, g) > δ and E x µ gx)χ S x) = 0 for S < d. These g and µ satisfy the conditions of Lemma 4.2, therefore we have disc,λ F g) 23) 2 d/2 where λ is obtained from µ as stated in Lemma 4.2 and l 2 2 )em/d. Since n = l m, 23) holds for n 2 2 )e) m d. It can be easily verified that Corr λ F f, F g ) = Corr µf, g) > δ. Thus, by plugging the value of disc,λ F g) in 6) of the generalized discrepancy method we get R ǫ F f ) d + logδ + 2ǫ ). 2 The desired result is obtained by applying Lemma Disjointness Separates BPP CC and NP CC As a corollary to our main theorem, we obtain the following lower bound for the Disjointness function. Corollary 5.3. for any constant ǫ > 0. R ǫ DISJ ) = Ω n )2 ) Proof. Let f = NOR m and f = NOR n. We now deg /3 NOR m ) = Θ m) by Theorem 2.2. Setting n = 2 2 )e deg /3 NOR m)) m, and writing 22) in terms of n gives the result for any constant ǫ > /6. The bound can be made to wor for every constant ǫ by a standard boosting argument. Observe that we get the same bound for the function G OR. It is not difficult to see that there is a Olog n) bit non-deterministic protocol for G OR and therefore this function separates the communication complexity classes BPP CC and NP CC for all = olog log n). 2

13 5.2 Other Symmetric Functions Theorem 5.2 does not immediately provide strong bounds on the communication complexity of G f for every symmetric f. For instance, if f is the MAJORITY function then one has to wor a little more to derive strong lower bounds. In this section, using the main result and Paturi s Theorem Theorem 2.2), we obtain a lower bound on the communication complexity of G f for each symmetric f. Let f : {0, }n {, } be the symmetric function induced from a predicate D : {0,,...,n} {, }. We denote by G D the function G f. For t {0,,...,n }, define D t : {0,,...,n t} {, } by D t i) = Di+t). Observe that the communication complexity of G D is at least the communication complexity of G Dt. Corollary 5.4. Let D : {0,,...,n} be any predicate with deg /3 D) = d. Let l 0 = l 0 D) and l = l D). Define T : N N by Then for any constant ǫ > 0, where n Tn) = 2 2 )e/d) ) R ǫ GD ) = Ω Ψl 0 ) + Tl ) ) 2 Ψl 0 ) = min{ω Tn)l 0 2 ), Ω Tn l 0 ) 2 ) }. Proof. There are three cases to consider. Case : Suppose l 0 Tn)/2. Let D : {0,,...,Tn)} {, } be such that for any z {0, } Tn), we have D z ) = D z ). By Theorem 5.2, the complexity of G D is Ωd/2 ) where d = deg /3 D ). By Paturi s Theorem, deg /3 D ) Tn)l 0 D ) = Tn)l 0 and so R ǫ GD ) = Ω Tn)l 0 2 ) Case 2: Suppose Tn)/2 < l 0 n/2. We find a lower bound on the communication complexity of G Dt where t = l 0 Tn l 0 )/2. Let D t : {0,,...,Tn l 0 )} {, } be such that D t z ) = D t z ). So by Theorem 5.2, the complexity of G Dt is Ωd/2 ) where d is the approximation degree of D t. We now D ttn l 0 )/2) = D t Tn l 0 )/2) = DTn l 0 )/2 + l 0 Tn l 0 )/2) = Dl 0 ) Dl 0 ) = D ttn l 0 )/2 ). Thus by Paturi s Theorem, deg /3 D t) Tn l 0 ) 2 /2. This implies R ǫ GD ) = Ω Tn l 0 ) 2 ). 3

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