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.