Babai, L., Frankl, P. and Simon, J. (1986): Complexity classes in communication complexity theory. In Proc. of 27th Ann. IEEE Symp. on Foundations of Comput. Sci., 337347.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....graph G = U, W; E) with IUI: IWI: m mo that satisfies one of propeies P1 or P2 with o(m ) placed by 5m must satisfy the other one with o(m ) replaced by em . Theorem A, either in full or in part, and its variants have appeared in several papers, because of its basic nature; see, for example, [3, 4, 5, 9, 11, 17, 47, 48] and the upper bound in Theorem 15.2 in [15] The importance of Theorem A comes from the fact that property P2 is in fact fundamental, as the next result shows. Let G = V, E) be a graph and U, W C V a pair of disjoint sets of vertices. Denote by E(U, W) the set of all edges between U and W, ....

....[ 4.2. A local condition for regularity. In this section, we state and prove a lemma that gives a sufficient condition for a possibly sparse bipartite graph to be z regular. The condition is local in nature. Similar results in somewhat different contexts have proved to be very useful; see [3, 4, 5, 9, 11, 17, 47, 48] and the proof of the upper bound in Theorem 15.2 in [15] due to J.H. Lindsey. 4.2.1. The statement of the lemma. Our lemma giving local conditions for the regularity of possibly sparse bipartite graphs is as follows. Lemma 15 (The local condition lemma) For all s 0 and all C 1, there is 5 ....

LAsz15 Babai, Pter Frankl, and Jinos Simon, Complexity classes in communication complexity theory (preliminary version), 27th Armual Symposium on Foundations of Computer Science (Toronto, Ontario, Canada), IEEE, 1986, pp. 337-347.

....protocol, and at the end of the protocol both of them must know the value of f . The communication complexity of f is defined to be the number of bits that need to be exchanged in the worst case by the best protocol for computing f . This model was extensively studied in the literature ( Yao79, BFS86, CG85, DGS84, FKN91, HR88, MS82, NW91, Raz90, AUY83] and many more) In this paper we also deal with boolean functions f(x 1 : xm ) where the partition of the input bits to the two players is not predefined. In this case we define the communication complexity of f as the maximum over all ....

....introduce the following notation for the communication complexity of families of functions. Definition 4 Let G be a set of functions, then we define R ffl (G) sup g2G R ffl (g) 6 2. 3 Known lower bounds A lower bound of Omega Gamma p n) for the randomized complexity of DISJ is given in [BFS86] this was improved to a tight Theta(n) in [KS87] and the proof was simplified in [Raz90] Theorem A. R(DISJ n ) Theta(n) A lower bound of Omega Gamma 22 n) for the randomized complexity of IP was proven in [Yao83] The bound was improved to Omega Gamma n= log n) in [Vaz87] and ....

L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th IEEE FOCS, pages 337--347, 1986.

....gap. Keywords: computational complexity, communication complexity, rank of a matrix, gap. 1 Introduction The notion of communication complexity was introduced by Yao [13] with the goal of providing a framework to analyze distributed computations. Since then it has attracted many scientists ([2, 3, 7, 8, 9, 10, 11]) Assume we want to compute a boolean function f with 2n variables split into two groups. Then we can view f as a function of two arguments f(x; y) where x and y are vectors each containing, e.g. half of the original variables. Furthermore we can interpret f as a matrix M where the rows are ....

L. Babai, P. Frankl, J. Simon, Complexity classes in communication complexity theory, Proc. 27-th IEEE FOCS, 1986, 337-347.

....the distribution is not necessarily a product distribution on . This is because # need not be a product distribution on X Y (although is the product of n copies of #) In fact, for set disjointness, it becomes essential to consider non product distributions to obtain an ## n) lower bound [BFS86] To handle this, we will use the fact that may be written as a convex combination = d#K # d d of product distributions d , where K is some index set. Such a decomposition, in general, is not unique, and we will choose one where the entropy of the collection of # d s, viewed as a ....

L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory (preliminary version). In Proceedings of the 27th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 337--347, 1986.

....rectangles labeled with the output of the protocol on that message sequence. Proving a lower bound on the number of rectangles needed in such a partition is then done by showing that all 1 correct rectangles are small. This approach or variants of it have been used by Yao [Y83] Babai et al. [BFS86], Razborov [R92] and adapted to partial functions also by Raz [R99] so that virtually all important lower bound proofs (except [KS92] for randomized communication complexity follow the described pattern. More precisely, the lower bound method (as described by Yao [Y83] goes as follows: First ....

....Corollary 1. MA(DISJ) n) while N( DISJ) O(log n) It seems unlikely that, but remains unknown whether DISJ has e cient AM protocols. Actually no separation between larger classes than MA 6= co MA is known within the communication complexity version of the polynomial hierarchy (see [BFS86], polynomial time is replaced by polylogarithmic communication in this de nition) Note that it is still open whether the polynomial hierarchy in communication complexity is strict. Actually we will give a lower bound in Theorem 7 showing that some explicit function is not contained in some even ....

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L. Babai, P. Frankl, J. Simon. Complexity classes in communication complexity theory. 27th IEEE Symp. Found. of Comp. Science, pp.303-312, 1986.

....showing N(ELIM(EQ ) Alice gets x 1 x 2 and Bob gets y 1 y 2 . Alice and Bob run the optimal nondeterministic protocol for ELIM(DISJ ) on (F (x 1 ) G(y k ) N, D(ELIM(INTER O(logn) Proof. This follows from Theorem 3.2 and Lemma 1.2. Note: Babai et al. [3] defined reductions between problems in communication complexity. The proof of Theorem 3.2 actually showed EQ cc DISJ, which enabled us to transfer our lower bound for ELIM(EQ ) to a lower bound for ELIM(DISJ ) Babai et al. 3] also defined P and NP , analogs of P and NP. Since we ....

....follows from Theorem 3.2 and Lemma 1.2. Note: Babai et al. 3] defined reductions between problems in communication complexity. The proof of Theorem 3. 2 actually showed EQ cc DISJ, which enabled us to transfer our lower bound for ELIM(EQ ) to a lower bound for ELIM(DISJ ) Babai et al. [3] also defined P and NP , analogs of P and NP. Since we have D(ELIM(NE n and D(ELIM(EQ n, and NE NP , EQ co NP , we can get lower bounds for any NP hard or coNP hard problem in communication complexity. We do this for graph properties in Section 4. Since the reductions ....

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory (preliminary version). In Proc. of the 27th IEEE Sym. on Found. of Comp. Sci., pages 337-- 347, 1986.

....the main argument that f has the randomness property required for proving the lower bound. This lemma is a stronger variant of a lemma that appears in [18] It is also interesting to compare this lemma to a result of Lindsey on the distribution of 1 s and 1 s in submatrices of Hadamard matrices [2, 12]. 8 Lemma 13 Let H = fh : I Og be a collection of universal hash functions. Let A I, B O and C H. Then, where p = jBj=jOj. Proof: De ne M h;x to be 1 if h(x) 2 B and 0 otherwise. Then, jProb x2A;h2C [h(x) 2 B] pj = jE h2C E x2A (M h;x p)j: Clearly, using Cauchy Schwartz ....

L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pages 337-347, 1986.

....DISJ n (D(s) s = 0) and the inner product function IP n (D(s) s (mod 2) The rank lower bound by Mehlhorn and Schmidt [MS82] immediately implies a tight n) lower bound on the deterministic communication complexity of both DISJ n and IP n . For the randomized algorithms, Vaz87, CG88, BFS86] proved an n) lower bound on the complexity of the inner product IP n , and [BFS86] also contained an n) lower bound for DISJ n . The latter bound was improved to the optimal n) in [KS92] and their proof was further simpli ed in [Raz92] The model of quantum communication complexity ....

....rank lower bound by Mehlhorn and Schmidt [MS82] immediately implies a tight n) lower bound on the deterministic communication complexity of both DISJ n and IP n . For the randomized algorithms, Vaz87, CG88, BFS86] proved an n) lower bound on the complexity of the inner product IP n , and [BFS86] also contained an n) lower bound for DISJ n . The latter bound was improved to the optimal n) in [KS92] and their proof was further simpli ed in [Raz92] The model of quantum communication complexity was also introduced by Yao [Yao93] Suppose that Alice and Bob can employ the laws of ....

L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th IEEE FOCS, pages 337-347, 1986.

....with legs Q 1 and Q 2 has (R ) R) 2. Observe that each element of Q j corresponds to an m subset of [N ] and that for any point x 2 R with ED(x) 1 the sets corresponding to the xQ 1 2 Q 1 and xQ 2 2 Q 2 must be disjoint. We will now apply an argument used by Babai, Frankl, and Simon [BFS86] to derive a lower bound on error communication complexity for this set disjointness problem. Note that the arguments used later by Kalayanasundaram and Schnitger [KS87] or Razborov [Raz90] to get optimal communication complexity bounds are not useful to us because these require precise linear ....

....Y , and Z but with Y = Q 1 and f(T ) 0 ] We obtain a set F of m subsets of [N ] such that jF j; jGj and for every T 2 F , 0 ] 4 1=6; i.e. each element of F intersects at most 4 1=6 of all elements of G. The following is a simple generalization of part of the argument in [BFS86] Proposition 6.21. Let d 3 and let F be a collection of m subsets of [N ] If jF j 2(4(d 1) d then F contains a sequence of p = dN= dm)e sets S 1 ; S p such that jS j S i j m=2 for j = 1; p, i.e. at least half the elements of S j do not occur in earlier sets. ....

Laszlo Babai, P. Frankl, and Janos Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science, pages 337--347, Toronto, Ontario, October 1986. IEEE.

....k # 0, 1 L k . Alice and Bob run the optimal nondeterministic protocol for ELIM(DISJ k ) on (F (x 1 ) F (x k ) G(y 1 ) G(y k ) Corollary 3.2. For all k, n # N, D(ELIM(INTER k ) # n O(logn) Proof. This follows from Theorem 3.2 and Lemma 1.2. Note: Babai et al. [3] defined reductions between problems in communication complexity. The proof of Theorem 3.2 actually showed EQ # cc DISJ, which enabled us to transfer our lower bound for ELIM(EQ k ) to a lower bound for ELIM(DISJ k ) Babai et al. 3] also defined P cc and NP cc , analogs of P and NP. ....

....from Theorem 3.2 and Lemma 1.2. Note: Babai et al. 3] defined reductions between problems in communication complexity. The proof of Theorem 3. 2 actually showed EQ # cc DISJ, which enabled us to transfer our lower bound for ELIM(EQ k ) to a lower bound for ELIM(DISJ k ) Babai et al. [3] also defined P cc and NP cc , analogs of P and NP. Since we have D(ELIM(NE k ) # n and D(ELIM(EQ k ) # n, and NE # NP cc , EQ # co NP cc , we can get lower bounds for any NP hard or coNP hard problem in communication complexity. We do this for graph properties in Section ....

[Article contains additional citation context not shown here]

L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory (preliminary version). In Proc. of the 27th IEEE Sym. on Found. of Comp. Sci., pages 337-- 347, 1986.

....n log n) qubits of communication. The question whether there exists a more e#cient quantum protocol for Disjoint is still an important open problem. This is especially relevant as 1840 HARRY BUHRMAN, RICHARD CLEVE, AND WIM VAN DAM Disjoint is a complete problem for the communication class co NP [2]. The same article [5] also contained the first exponential separation for the exact distributed computation of a partial two party function that is related to the Deutsch Jozsa problem of [11] In [27] Raz improved on these results by establishing an exponential separation between classical and ....

L. Babai, P.G. Frankl, and J. Simon, Complexity classes in communication complexity theory, in Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, 1986, pp. 337--347.

....In subsection 1 we present a fairly simple function which is complete for the class of boolean functions whose 1 round probabilistic communication complexity is polylog(n) First we need to define completeness in this context. This is done using rectangular reductions, which were introduced in [1]. ffl In subsection 2 we shall use the definitions of subsection 1 for the quantum case and define a complete problem for the class of boolean functions whose 1 round quantum communication complexity is polylog(n) ffl In subsection 3 we shall define another complete problem for the quantum ....

L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science, pages 337--347, 1986.

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Babai, L., Frankl, P. and Simon, J. (1986): Complexity classes in communication complexity theory. In Proc. of 27th Ann. IEEE Symp. on Foundations of Comput. Sci., 337347.

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. 27th IEEE Symposium on Foundations of Computer Science, pp. 303-312, 1986.

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L. Babai, P. Frankl, J. Simon. Complexity classes in communication complexity theory. 27th IEEE Symposium Foundations of Computer Science, pp. 303--312, 1986.

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science, pages 337--347, Los Angeles, Ca., USA, Oct. 1986. IEEE Computer Society Press.

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science, pages 337--347, Los Angeles, Ca., USA, Oct. 1986. IEEE Computer Society Press.

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Laszlo Babai, P. Frankl, and Janos Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science, pages 337-347, Toronto, Ontario, October 1986. IEEE.

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In IEEE Symposium on Foundations of Computer Science, pages 337--347, 1986.

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of 27th IEEE FOCS, pages 337--347, 1986.

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L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of 27th IEEE FOCS, pages 337-347, 1986.

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Laszlo Babai, Peter Frankl, and Janos Simon, Complexity classes in communication complexity theory (preliminary version), 27th Annual Symposium on Foundations of Computer Science (Toronto, Ontario, Canada), IEEE, 1986, pp. 337-347. 1

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Babai, L., P. Frankl, and J. Simon. "Complexity Classes in Communication Complexity Theory," Proceedings of FOCS, 337--347, 1986.

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Babai, L., P. Frankl, and J. Simon. \Complexity Classes in Communication Complexity Theory," Proceedings of FOCS, 337-347, 1986.

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Babai, L., P. Frankl, and J. Simon. "Complexity Classes in Communication Complexity Theory," Proceedings of FOCS, 337---347, #986.

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