New bounds on classical and quantum one-way communication complexity

In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical one-way model, our bound extends the well known upper bound of Kremer, Nisan and Ron [I. Kremer, N. Nisan, D. Ron, On randomized one-round communication complexity, in: Proceedings of The 27th ACM Symposium on Theory of Computing, STOC, 1995, pp. 596-605] to include non-product distributions. Let @[email protected]?(0,1/2) be a constant. We show that for a boolean function f:XxY->{0,1} and a non-product distribution @m on XxY, D"@e^1^,^@m(f)=O((I(X:Y)+1)@?VC(f)), where D"@e^1^,^@m(f) represents the one-way distributional communication complexity of f with error at most @e under @m; VC(f) represents the Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual information, under @m, between the random inputs of the two parties. For a non-boolean function f:XxY->{1,...,k} (k>=2 an integer), we show a similar upper bound on D"@e^1^,^@m(f) in terms of k,I(X:Y) and the pseudo-dimension of f^'=deffk, a generalization of the VC-dimension for non-boolean functions. In the quantum one-way model we provide a lower bound on the distributional communication complexity, under product distributions, of a function f, in terms of the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f. We show for a non-boolean total function f:XxY->Z and a product distribution @m on XxY, Q"@e"^"3"/"8^1^,^@m(f)[email protected](rec"@e^1^,^@m(f)), where Q"@e"^"3"/"8^1^,^@m(f) represents the quantum one-way distributional communication complexity of f with error at most @e^3/8 under @m and rec"@e^1^,^@m(f) represents the one-way rectangle bound of f with error at most @e under @m. Similarly for a non-boolean partial function f:XxY->[email protected]?{*} and a product distribution @m on XxY, we show, Q"@e"^"6"/"("2"@?"1"5"^"4")^1^,^@m(f)[email protected](rec"@e^1^,^@m(f)).

[1]  Philip M. Long,et al.  Fat-shattering and the learnability of real-valued functions , 1994, COLT '94.

[2]  Jaikumar Radhakrishnan,et al.  The Communication Complexity of Correlation , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[3]  David Haussler,et al.  Learnability and the Vapnik-Chervonenkis dimension , 1989, JACM.

[4]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[5]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[6]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[7]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[8]  Scott Aaronson,et al.  The learnability of quantum states , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  Ashwin Nayak,et al.  Optimal lower bounds for quantum automata and random access codes , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[10]  Alexander A. Razborov,et al.  On the Distributional Complexity of Disjointness , 1992, Theor. Comput. Sci..

[11]  Robert König,et al.  Universally Composable Privacy Amplification Against Quantum Adversaries , 2004, TCC.

[12]  Noam Nisan,et al.  On Randomized One-round Communication Complexity , 1995, STOC '95.

[13]  Ran Raz,et al.  Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography , 2008, SIAM J. Comput..

[14]  E. Kushilevitz,et al.  Communication Complexity: Basics , 1996 .

[15]  Ilan Newman,et al.  Private vs. Common Random Bits in Communication Complexity , 1991, Inf. Process. Lett..

[16]  Norbert Sauer,et al.  On the Density of Families of Sets , 1972, J. Comb. Theory A.

[17]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[18]  Thomas M. Cover,et al.  Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) , 2006 .

[19]  Ueli Maurer,et al.  On the power of quantum memory , 2005, IEEE Transactions on Information Theory.

[20]  Barbara M. Terhal,et al.  The Bounded-Storage Model in the Presence of a Quantum Adversary , 2008, IEEE Transactions on Information Theory.

[21]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[22]  Hartmut Klauck,et al.  Direct Product Theorems for Communication Complexity via Subdistribution Bounds , 2007, Electron. Colloquium Comput. Complex..

[23]  S. Muthukrishnan,et al.  Data streams: algorithms and applications , 2005, SODA '03.