Communication Complexity Of Boolean Function Research Articles

Compressibility of Positive Semidefinite Factorizations and Quantum Models

We investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products. This can either be regarded as compression of positive semidefinite factorizations of nonnegative matrices or (if the matrices are subject to additional normalization constraints) as compression of quantum models. We derive both lower and upper bounds on compressibility. Applications are broad and range from the statistical analysis of experimental data to bounding the one-way quantum communication complexity of Boolean functions.

Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions

The two-party communication complexity of Boolean function f is known to be at least log rank (M f ), i.e., the logarithm of the rank of the communication matrix of f [19]. Lovasz and Saks [17] asked whether the communication complexity of f can be bounded from above by (log rank (M f )) c , for some constant c . The question was answered affirmatively for a special class of functions f in [17], and Nisan and Wigderson proved nice results related to this problem [20], but, for arbitrary f , it remained a difficult open problem. We prove here an analogous polylogarithmic upper bound in the stronger multiparty communication model of Chandra et al. [6], which, instead of the rank of the communication matrix, depends on the L 1 norm of function f , for arbitrary Boolean function f .

Lower Bounds on the Multiparty Communication Complexity

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 functionf(x1, …, xn) was derived, namely of the order (k0C0)(k1C1)2, up to logarithmic factors, wherek1andC1are the number of processors accessed and the bits exchanged in a nondeterministic algorithm forf, andk0andC0are the analogous parameters for the complementary function 1−f. We show thatC0⩽n(1+2C1) andD⩽n(1+2C1), whereDis the number of bits exchanged by a deterministic algorithm computingf. 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.

A comparison of two lower-bound methods for communication complexity

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. 1. (i) For almost all Boolean functions of 2 n 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 2 n + log 2 10. A specific sequence {ƒ 2n} n = 1 ∞ of Boolean functions, where ƒ 2n has 2 n variables, is constructed such that the Rank method provides exponentially higher lower bounds for ƒ 2n than the Fooling Set method. 2. (ii) A specific sequence { h 2 n } n = 1 ∞ of Boolean functions is constructed such that the Fooling Set method provides a lower bound of n for h 2 n , whereas the Rank method provides only (log 2 3) 2 · n ≈ 0.79 · n as a lower bound. 3. (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 Aho et al. (1983). 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.