A direct sum theorem for corruption and the multiparty NOF communication complexity of set disjointness

We prove that corruption, one of the most powerful measures used to analyze 2-party randomized communication complexity, satisfies a strong direct sum property under rectangular distributions. This direct sum bound holds even when the error is allowed to be exponentially close to 1. We use this to analyze the complexity of the widely-studied set disjointness problem in the usual "number-on-the-forehead" (NOF) model of multiparty communication complexity.

[1]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[2]  Ran Raz,et al.  A direct product theorem , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[3]  Andrew Chi-Chih Yao,et al.  Informational complexity and the direct sum problem for simultaneous message complexity , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[4]  Noga Alon,et al.  The space complexity of approximating the frequency moments , 1996, STOC '96.

[5]  Noam Nisan,et al.  Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs , 1992, J. Comput. Syst. Sci..

[6]  Ran Raz,et al.  Super-logarithmic depth lower bounds via the direct sum in communication complexity , 1995, computational complexity.

[7]  Hartmut Klauck,et al.  Quantum and classical strong direct product theorems and optimal time-space tradeoffs , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[8]  Richard J. Lipton,et al.  Multi-party protocols , 1983, STOC.

[9]  Fan Chung Graham,et al.  Communication Complexity and Quasi Randomness , 1993, SIAM J. Discret. Math..

[10]  Ran Raz,et al.  Super-logarithmic depth lower bounds via direct sum in communication complexity , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[11]  Eyal Kushilevitz,et al.  Fractional covers and communication complexity , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[12]  Ziv Bar-Yossef,et al.  An information statistics approach to data stream and communication complexity , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[13]  Bala Kalyanasundaram,et al.  The Probabilistic Communication Complexity of Set Intersection , 1992, SIAM J. Discret. Math..

[14]  Ronen Shaltiel.  Towards proving strong direct product theorems , 2003, computational complexity.

[15]  Jaikumar Radhakrishnan,et al.  A Direct Sum Theorem in Communication Complexity via Message Compression , 2003, ICALP.

[16]  Ran Raz,et al.  The BNS-Chung criterion for multi-party communication complexity , 2000, computational complexity.

[17]  Michael E. Saks,et al.  Space lower bounds for distance approximation in the data stream model , 2002, STOC '02.

[18]  Satyanarayana V. Lokam,et al.  Communication Complexity of Simultaneous Messages , 2003, SIAM J. Comput..

[19]  Subhash Khot,et al.  Near-optimal lower bounds on the multi-party communication complexity of set disjointness , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[20]  Noam Nisan,et al.  On Yao's XOR-Lemma , 1995, Electron. Colloquium Comput. Complex..

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

[22]  Thomas P. Hayes,et al.  The Cost of the Missing Bit: Communication Complexity with Help , 1998, STOC '98.

[23]  Michael E. Saks,et al.  Products and help bits in decision trees , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[24]  Hartmut Klauck,et al.  Rectangle size bounds and threshold covers in communication complexity , 2002, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[25]  Noam Nisan,et al.  Rounds in communication complexity revisited , 1991, STOC '91.

[26]  Jin-Yi Cai.  Lower bounds for constant depth circuits in the presence of help bits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[27]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[28]  Ziv Bar-Yossef,et al.  Information theory methods in communication complexity , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.