Non-Deterministic Communication Complexity with Few Witnesses

2007

We solve some fundamental problems in the number-on-forehead (NOF) k-party communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a one-sided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for k= n O (1) players, where n is the number of bits on each players' forehead. This separates the analogues of RP and P in the NOF communication model.

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.

2003

Deterministic, probabilistic, nondeterministic, and alternating complexity classes defined by polylogarithmic communication are considered. Main results are (1) extending work of Ja'Ja', Prasanna Kumar, and Simon, we give a simple technique allowing translation of most known separation and containment results for complexity classes of the fixed partition model to the more difficult optimal partition model, where few results were previously known; and (2) demonstration that a certain natural language (block-equality) in Zy is also, unexpectedly, in flee 2 .

Journal of Computer and System Sciences, 1990

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.

Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87

Improving a result of Mehlhorn and Schmidt, a function f with deterministiccommunication complexity n2 is shown to have Las Vegas communication complexity O(n:l. This is the best possible, because the deterministic complexity cannot be more than the square of the Las Vegas communication complexity for any function.

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.

Journal of Cryptology, 1993

The fact that there are zero-knowledge proofs for all languages in NP (see , , and [5]) has, potentially, enormous implications to cryptography. For cryptographers, the issue is no longer "which languages in NP have zeroknowledge proofs" but rather "which languages in NP have practical zeroknowledge proofs." Thus, the concrete complexity of zero-knowledge proofs for different languages must be established.

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.

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.