Abstract
The fooling pairs method is one of the standard methods for proving lower bounds for deterministic two-player communication complexity. We study fooling pairs in the context of randomized communication complexity. We show that every fooling pair induces far away distributions on transcripts of private-coin protocols. We use the above to conclude that the private-coin randomized \(\varepsilon \)-error communication complexity of a function f with a fooling set \(\mathcal S\) is at least order \(\log \frac{\log |\mathcal S|}{\varepsilon }\). This relationship was earlier known to hold only for constant values of \(\varepsilon \). The bound we prove is tight, for example, for the equality and greater-than functions.
As an application, we exhibit the following dichotomy: for every boolean function f and integer n, the (1/3)-error public-coin randomized communication complexity of the function \(\bigvee _{i=1}^{n}f(x_i,y_i)\) is either at most c or at least n/c, where \(c>0\) is a universal constant.
M. Sinha—Partially supported by BSF.
A. Yehudayoff—Horev fellow – supported by the Taub foundation. Supported by ISF and BSF.
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Notes
- 1.
\(R\subseteq \mathcal {X}\times \mathcal Y\) is an f-monochromatic rectangle if \(R=A\times B\) for some \(A\subseteq \mathcal {X}, B\subseteq \mathcal Y\) and f is constant over R.
- 2.
- 3.
We may assume that say \(\varepsilon < 2^{-12}\) by repeating the given randomized protocol a constant number of times.
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Moran, S., Sinha, M., Yehudayoff, A. (2016). Fooling Pairs in Randomized Communication Complexity. In: Suomela, J. (eds) Structural Information and Communication Complexity. SIROCCO 2016. Lecture Notes in Computer Science(), vol 9988. Springer, Cham. https://doi.org/10.1007/978-3-319-48314-6_4
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