Probabilistic Communication Complexity Over The Reals

Abstract.Deterministic and probabilistic communication protocols are introduced in which parties can exchange the values of polynomials (rather than bits in the usual setting). It is established a sharp lower bound 2n on the communication complexity of recognizing the 2n-dimensional orthant, on the other hand the probabilistic communication complexity of recognizing it does not exceed 4. A polyhedron and a union of hyperplanes are constructed in $$\mathbb{R}^{2n}$$ for which a lower bound n on the probabilistic communication complexity of recognizing each is proved. As a consequence this bound holds also for the EMPTINESS and the KNAPSACK problems.

[1]  Farid M. Ablayev,et al.  A Discrete Approximation and Communication Complexity Approach to the Superposition Problem , 2001, FCT.

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

[3]  Markus Bläser,et al.  Algebraic Communication Complexity of Distributed Computation , 2005 .

[4]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[5]  Harold Abelson,et al.  Lower bounds on information transfer in distributed computations , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[6]  Ronald de Wolf,et al.  Communication complexity lower bounds by polynomials , 1999, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[7]  Marek Karpinski,et al.  Randomization and the computational power of analytic and algebraic decision trees , 1996, computational complexity.

[8]  Jan Kraíček,et al.  Interpolation by a Game , 1998 .

[9]  Dima Grigoriev,et al.  Randomized Complexity Lower Bound for Arrangements and Polyhedra , 1999, Discret. Comput. Geom..

[10]  Friedhelm Meyer auf der Heide,et al.  A lower bound for randomized algebraic decision trees , 1996, STOC '96.

[11]  Gabriel Ciobanu,et al.  Fundamentals of Computation Theory , 2003, Lecture Notes in Computer Science.

[12]  Peter Bürgisser,et al.  Completeness and Reduction in Algebraic Complexity Theory , 2000, Algorithms and computation in mathematics.

[13]  Dima Grigoriev,et al.  Solving Systems of Polynomial Inequalities in Subexponential Time , 1988, J. Symb. Comput..

[14]  Friedhelm Meyer auf der Heide,et al.  Simulating Probabilistic by Deterministic Algebraic Computation Trees , 1985, Theor. Comput. Sci..

[15]  Jan Krajícek.  Interpolation by a Game , 1998, Math. Log. Q..

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