Communication Lower Bounds Using Directional Derivatives

We study the set disjointness problem in the most powerful model of bounded-error communication, the k-party randomized number-on-the-forehead model. We show that set disjointness requires Ω(√n/2kk) bits of communication, where n is the size of the universe. Our lower bound generalizes to quantum communication, where it is essentially optimal. Proving this bound was a longstanding open problem even in restricted settings, such as one-way classical protocols with k=4 parties [Wigderson 1997]. The proof contributes a novel technique for lower bounds on multiparty communication, based on directional derivatives of protocols over the reals.

[1]  Ronald de Wolf,et al.  A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[2]  Shachar Lovett,et al.  The Complexity of Boolean Functions in Different Characteristics , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[3]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[4]  Emanuele Viola,et al.  One-way multiparty communication lower bound for pointer jumping with applications , 2009, Comb..

[5]  Alexander A. Sherstov.  SeparatingAC0 from Depth-2 Majority Circuits , 2009, SIAM J. Comput..

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

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

[8]  A. N. Alekseichuk.  On Complexity of Computation of Partial Derivatives of Boolean Functions Realized by Zhegalkin Polynomials , 2001 .

[9]  Ben Green,et al.  Finite field models in additive combinatories , 2004, BCC.

[10]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[11]  David Steurer,et al.  Rounding Parallel Repetitions of Unique Games , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[12]  Hartmut Klauck,et al.  Direct product theorems for classical communication complexity via subdistribution bounds: extended abstract , 2008, STOC.

[13]  Hartmut Klauck,et al.  Lower bounds for quantum communication complexity , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

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

[15]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[16]  Alexander A. Sherstov.  Separating AC0 from depth-2 majority circuits , 2007, STOC '07.

[17]  Arkadev Chattopadhyay.  Circuits, communication and polynomials , 2009 .

[18]  Troy Lee,et al.  Disjointness is Hard in the Multiparty Number-on-the-Forehead Model , 2007, 2008 23rd Annual IEEE Conference on Computational Complexity.

[19]  Avi Wigderson,et al.  A Strong Direct Product Theorem for Corruption and the Multiparty Communication Complexity of Disjointness , 2006, computational complexity.

[20]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[21]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[22]  Eyal Kushilevitz,et al.  Communication Complexity: Index of Notation , 1996 .

[23]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[24]  Adi Shraibman,et al.  Lower bounds in communication complexity based on factorization norms , 2009 .

[25]  T. Cusick,et al.  Bent Boolean functions , 2009 .

[26]  Ronald de Wolf,et al.  Bounds for small-error and zero-error quantum algorithms , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[27]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[28]  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..

[29]  A. Razborov.  Quantum communication complexity of symmetric predicates , 2002, quant-ph/0204025.

[30]  Andrew Chi-Chih Yao,et al.  Theory and application of trapdoor functions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[31]  Arkadev Chattopadhyay,et al.  Multiparty Communication Complexity of Disjointness , 2008, Electron. Colloquium Comput. Complex..

[32]  David Pollard,et al.  A User's Guide to Measure Theoretic Probability by David Pollard , 2001 .

[33]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[34]  Johan Håstad,et al.  On the power of small-depth threshold circuits , 1991, computational complexity.

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

[36]  Avi Wigderson,et al.  Quantum vs. classical communication and computation , 1998, STOC '98.

[37]  Hartmut Klauck.  A strong direct product theorem for disjointness , 2010, STOC '10.

[38]  Paul Beame,et al.  MULTIPARTY COMMUNICATION COMPLEXITY AND THRESHOLD CIRCUIT SIZE OF AC0∗ , 2010 .

[39]  Alexander A. Sherstov.  The multiparty communication complexity of set disjointness , 2012, STOC '12.

[40]  Alexander A. Sherstov.  The Pattern Matrix Method , 2009, SIAM J. Comput..

[41]  Pantelimon Stanica,et al.  Cryptographic Boolean Functions and Applications , 2009 .

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

[43]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[44]  Ángel Martín del Rey,et al.  On the boolean partial derivatives and their composition , 2012, Appl. Math. Lett..

[45]  Vince Grolmusz,et al.  The BNS Lower Bound for Multi-Party Protocols in Nearly Optimal , 1994, Inf. Comput..

[46]  T. N. Sriram.  Asymptotics in Statistics–Some Basic Concepts , 2002 .

[47]  Alexander A. Sherstov.  Strong direct product theorems for quantum communication and query complexity , 2010, STOC '11.

[48]  Yaoyun Shi,et al.  Quantum communication complexity of block-composed functions , 2007, Quantum Inf. Comput..

[49]  Alexander A. Sherstov.  The Multiparty Communication Complexity of Set Disjointness , 2016, SIAM J. Comput..

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

[51]  A. Chattopadhyay.  Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits , 2007, FOCS 2007.

[52]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[53]  Eric Allender,et al.  A note on the power of threshold circuits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[54]  Dmitry Gavinsky,et al.  A Separation of NP and coNP in Multiparty Communication Complexity , 2010, Theory Comput..

[55]  Ran Raz,et al.  A Counterexample to Strong Parallel Repetition , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[56]  Gérard Y. Vichniac,et al.  Boolean derivatives on cellular automata , 1991 .

[57]  Harry Buhrman,et al.  On Computation and Communication with Small Bias , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[58]  Denis Thérien,et al.  Computational complexity questions related to finite monoids and semigroups , 2003 .

[59]  D. Pollard.  A User's Guide to Measure Theoretic Probability by David Pollard , 2001 .

[60]  Grace L. Yang,et al.  Asymptotics In Statistics , 1990 .

[61]  Arkadev Chattopadhyay.  Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[62]  Gideon Schechtman,et al.  Lower Bounds on Quantum Multiparty Communication Complexity , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[63]  Thomas Vidick,et al.  MULTIPLAYER XOR GAMES AND QUANTUM COMMUNICATION COMPLEXITY WITH CLIQUE-WISE ENTANGLEMENT , 2009, 0911.4007.

[64]  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.