Communication Complexity of Simultaneous Messages (original) (raw)
Lower Bounds on the Multiparty Communication Complexity
Journal of Computer and System Sciences, 1998
We derive a general technique for obtaining lower bounds on the multiparty communication complexity of boolean functions. We extend the two-party method based on a crossing sequence argument introduced by Yao to the multiparty communication model. We use our technique to derive optimal lower and upper bounds of some simple boolean functions. Lower bounds for the multiparty model have been a challenge since (D. Dolev and T. Feder, in``Proceedings, 30th IEEE FOCS, 1989,'' pp. 428 433), where only an upper bound on the number of bits exchanged by a deterministic algorithm computing a boolean function f (x 1 , ..., x n) was derived, namely of the order (k 0 C 0)(k 1 C 1) 2 , up to logarithmic factors, where k 1 and C 1 are the number of processors accessed and the bits exchanged in a nondeterministic algorithm for f, and k 0 and C 0 are the analogous parameters for the complementary function 1& f. We show that C 0 n(1+2 C1) and D n(1+2 C1), where D is the number of bits exchanged by a deterministic algorithm computing f. We also investigate the power of a restricted multiparty communication model in which the coordinator is allowed to send at most one message to each party.
SIAM Journal on Computing, 2012
We prove an n Ω(1) /4 k lower bound on the randomized k-party communication complexity of depth 4 AC 0 functions in the number-on-forehead (NOF) model for up to Θ(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC 0 function for ω(log log n) players. For non-constant k the bounds are larger than all previous lower bounds for any AC 0 function even for simultaneous communication complexity.
Multiparty communication complexity and very hard functions
Information and Computation, 2004
A boolean function f(x 1 ,. .. , x n) with x i ∈ {0, 1} m for each i is hard if its nondeterministic multiparty communication complexity (introduced in [in: Proceedings of the 30th IEEE FOCS, 1989, p. 428-433]), C(f), is at least nm. Note that C(f) nm for each f(x 1 ,. .. , x n) with x i ∈ {0, 1} m for each i. A boolean function is very hard if it is hard and its complementary function is also hard. In this paper, we show that randomly chosen boolean function f(x 1 ,. .. , x n) with x i ∈ {0, 1} m for each i is very hard with very high probability (for n 3 and m large enough). In [in: Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science, LNCS 900, 1995, p. 350-360], it has been shown that if f(x 1 ,. .. , x k ,. .. , x n) = f 1 (x 1 ,. .. , x k) • f 2 (x k+1 ,. .. , x n), where C(f 1) > 0 and C(f 2) > 0, then C(f) = C(f 1) + C(f 2). We prove here an analogical result: If f(x 1 ,. .. , x k ,. .. , x n) = f 1 (x 1 ,. .. , x k) ⊕ f 2 (x k+1 ,. .. , x n) then DC(f) = DC(f 1) + DC(f 2), where DC(g) denotes the deterministic multiparty communication complexity of the function g and "⊕" denotes the parity function.
A comparison of two lower-bound methods for communication complexity
Theoretical Computer Science, 1996
The methods "Rank" and "Fooling Set" for proving lower bounds on the deterministic communication complexity of Boolean functions are compared. The main results are as follows. (i) For almost all Boolean functions of 2n variables the Rank method provides the lower bound n on communication complexity, whereas the Fooling Set method provides only the lower bound d(n) < log, n+log, 10. A specific sequence {fin},OO=, of Boolean functions, where f;, has 2n variables, is constructed such that the Rank method provides exponentially higher lower bounds for fz,, than the Fooling Set method. (ii) A specific sequence {/z~~}Z, of Boolean functions is constructed such that the Fooling Set method provides a lower bound of n for hzn, whereas the Rank method provides only (log, 3)/2 n M 0.79. n as a lower bound. (iii) It is proved that lower bounds obtained by the Fooling Set method are better by at most a factor of two compared with lower bounds obtained by the Rank method. These three results together solve the last problem about the comparison of lower bound methods on communication complexity left open in . Finally, it is shown that an extension of the Fooling Set method provides lower bounds that are tight (up to a polynomial) for all Boolean functions.
The Multiparty Communication Complexity of Exact-T: Improved Bounds and New Problems
Lecture Notes in Computer Science, 2006
Let x1, . . . , x k be n-bit numbers and T ∈ N. Assume that P1, . . . , P k are players such that Pi knows all of the numbers except xi. They want to determine if k j=1 xj = T by broadcasting as few bits as possible. In an upper bound of O( √ n) bits was obtained for the k = 3 case, and a lower bound of ω(1) for k ≥ 3 when T = Θ(2 n ). We obtain (1) for k ≥ 3 an upper bound of k +O((n+log k) 1/( lg(2k−2) ) ), (2) for k = 3, T = Θ(2 n ), a lower bound of Ω(log log n), (3) a generalization of the protocol to abelian groups, (4) lower bounds on the multiparty communication complexity of some regular languages, and (5) empirical. results for k = 3,
Communication complexity towards lower bounds on circuit depth
2002
Karchmer, Raz, and Wigderson, 1991, discuss the circuit depth complexity of n bit Boolean functions constructed by composing up to d = log n= log log n levels of k = log n bit boolean functions. Any such function is in AC 1. They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires dk 2 (log 2 n= log log n) depth. This would separate AC 1 from NC 1. They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk O(d 2 (k log k) 1=2) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring (k) circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds.
Languages with Bounded Multiparty Communication Complexity
Research supported in part by the NFS (M. Szegedy), NSERC (A. Chattopadhyay, M. Koucký, P. Tesson, D. Thérien), FQRNT (M. Koucký, D. Thérien) and the Alexander von Humboldt Foundation (P. Tesson and D. Thérien). We are also grateful to Pavel Pudlák for suggesting the use of the Hales-Jewett Theorem.
Separating deterministic from randomized multiparty communication complexity
2010
Abstract: We solve some fundamental problems in the number-on-forehead (NOF) kplayer communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with one-sided falsepositives error probability of 1/3, but which has linear communication complexity for deterministic protocols and, in fact, even for the more powerful nondeterministic protocols.
The linear-array problem in communication complexity resolved
Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97
considered the following seenario: k+ 1 processors Po, . . ..P~. connected by k links to form a linear array, are to compute a function .f(z, y), x ~X, y c Y, on a finite domain X x Y, where x is only known to PO, y is only known to Pk; the intermediate processors Pl, . . . . Pk_l do not have any information. The processors compute f(x, y) by exchanging binary messages across the links, according to some protocol @. Let 11~(~) denote the minimal complexity of such a protocol 0, i. e., the total number of bits sent across all links for the worst case input, and let ~(~) = D1 (f ) denote the (standard) 2-party communication complexity off. Tiwari proved that Dk(~) ~k ~(D(f) -O(l)) for almost all functions $ and conjectured this inequality to be true for all $. His conjecture was falsified by : they exhibited a function f for which Dk (f ) is essentially bounded above by ~kD( f ). The best general lower bound known ing machines can be obtained directly by considering the deterministic two-party communication complexity of its restrictions f 1{.,1}2., for n z 1.