M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via direct sum in communication complexity. In 6th Structure in Complexity Theory Conference, pages 299-304, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....function. For example, the two party set disjointness function can be written in terms of n two bit and functions, one for each coordinate. By computing each and function separately, we trivially obtain a protocol to compute disjointness. The direct sum question for communication protocols [KRW95] asks whether there is a protocol with considerably less communication. We consider a related question, namely, the direct sum property for the information content of the transcripts For a finite sequence a = a 1 , a 2 , where each element belongs to [n] and for j [n] let f j (a) ....

Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3/4):191--204, 1995.

....literature does not seem to have a formal statement. Before making a formal statement we need to adapt some conventions. Convention: A function f : 1 is actually a family of functions, one for each n. We think of n as growing. We take the following formal statement which is implicit in [29] to be DSC: Direct Sum Conjecture (DSC) If f : 1 then D(f ) k(D(f) O(1) Formally (#N ) #K) #c) #n N ) #k K) D(f k(D(f) c) DSC is interesting for two reasons. 1) It is quite natural to compare solving k problems seperately to solving them together. The complexity ....

.... of a problem has been looked at in a variety of fields including decision trees [9, 40] computability [7, 22] complexity [2, 10, 11, 31] straightline programs [15, 14, 21, 52] and circuits [43] 2) This conjecture arose in the study of circuits since a variant of it implies NC (see [29, 28] for connections to circuits, and see [34, Pages 42 48] for a more recent discussion) The reasons for the form ) k(D(f) O(1) are (a) the counterexample above still satisfies D(f k(D(f) O(1) and (b) the variant needed for NC allows for an additive constant. While there are ....

M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5, 1995. Earlier version in STRUCTURES 1991.

....to have a formal statement. Before making a formal statement we need to adapt some conventions. Convention: A function f : 0, 1 n 0, 1 n # 0, 1 is actually a family of functions, one for each n. We think of n as growing. We take the following formal statement which is implicit in [29] to be DSC: Direct Sum Conjecture (DSC) If f : 0, 1 n 0, 1 n # 0, 1 then D(f k ) k(D(f) O(1) Formally (#N ) #K) #c) #n # N ) #k # K) D(f k ) # k(D(f) c) DSC is interesting for two reasons. 1) It is quite natural to compare solving k problems seperately to ....

.... problem has been looked at in a variety of fields including decision trees [9, 40] computability [7, 22] complexity [2, 10, 11, 31] straightline programs [15, 14, 21, 52] and circuits [43] 2) This conjecture arose in the study of circuits since a variant of it implies NC 1 #= NC 2 (see [29, 28] for connections to circuits, and see [34, Pages 42 48] for a more recent discussion) The reasons for the form D(f k ) k(D(f) O(1) are (a) the counterexample above still satisfies D(f k ) # k(D(f) O(1) and (b) the variant needed for NC 1 #= NC 2 allows for an additive ....

M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5,

....Parameters like the rank and the independence number which yield further lower bounds on the communication complexity are multiplicative under the Kronecker product. A further line of research leading to direct sum methods in communication complexity goes back to Karchmer, Raz, and Wigderson [25], cf. also [28] pp. 42 48. Their question was if it is easier to solve communication problems simultaneously than separately. Recall the definition of a vector valued function f n ( x 1 , xn ) y 1 , yn ) f(x 1 , y 1 ) f(xn , yn ) An obvious upper bound on the ....

....si, which would cost 2n bits. So the amortized communication complexity of the function si is 1 n lim n## C(si n ) log 3. Further with Theorem 2 it is also clear that this is the maximum compression for basic Boolean functions f : 0, 1 0, 1 # 0, 1 . Karchmer, Raz, and Wigderson [25] asked how much better simultaneous computations are compared to the componentwise evaluation of the function f n for basic functions f : 0, 1 m 0, 1 m # 0, 1 . They conjectured that the amortized communication complexity 1 n lim sup n## C(f n ) cannot di#er from C(f) by more ....

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M. Karchmer, R. Raz, and A. Wigderson, "Super-logarithmic depth lower bounds via direct sum methods in communication complexity", Proc. 6th IEEE Structure in Complexity Theory, 1991, 299 - 304

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M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via direct sum in communication complexity. In 6th Structure in Complexity Theory Conference, pages 299-304, 1991.

....n s4 s jj . Corollary 2. Any depth 3 threshold circuit which computes GIP2 n;log n and has bottom fan in at most 1 3 log n, must be of size exp i n Omega Gamma30 j . We define the following function: f n (x) n M i=1 log n j=1 n M k=1 x ijk : 1) Note that in the notation of [KRW91], f n is the composition of GIP2 and PARITY functions, namely f n j GIP2 n;log n ffi PARITY n . The main result of this note is the following Theorem 3. Any depth 3 threshold circuit which computes f n and has AND gates at the bottom must be of size n Omega Gamma1 6 n) Remark 4. In fact, ....

M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via direct sum in communication complexity. In 6th Structure in Complexity Theory Conference, pages 299--304, 1991.

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M. Karchmer, R. Raz, and A. Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Comput. Complexity, 5(3/4):191-- 204, 1995.

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