An Improved Communication-Randomness Tradeoff

Abstract

Two processors receive inputs X and Y respectively. The communication complexity of the function f is the number of bits (as a function of the input size) that the processors have to exchange to compute f(X,Y) for worst case inputs X and Y. The List-Non-Disjointness problem (X=(x ¹,...,x ⁿ), Y=(y ¹,...,y ⁿ), $x^{j},y^{j}\in {\rm Z}^{n}_{2}$, to decide whether $\exists_{j}x^{j}=y^{j}$) exhibits maximal discrepancy between deterministic n ² and Las Vegas (Θ(n)) communication complexity. Fleischer, Jung, Mehlhorn (1995) have shown that if a Las Vegas algorithm expects to communicate Ω(n logn) bits, then this can be done with a small number of coin tosses.

Even with an improved randomness efficiency, this result is extended to the (much more interesting) case of efficient algorithms (i.e. with linear communication complexity). For any R ∈ ℕ, R coin tosses are sufficient for O(n+n ² /2^R) transmitted bits.