New Strong Direct Product Results in Communication Complexity
Rahul Jain · Journal of the ACM · 2015
We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and ϵ > 0 be a constant. Let R 1,pub ϵ ( f ) represent the communication complexity of f , with worst-case error ϵ in this model. We show that if for computing f k ( k independent copies of f ) in this model, o ( k ċ R 1, pub 1/3 ( f )) communication is used, then the success is exponentially small in k . We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest. Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω( n ) for the set-disjointness problem on n -bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.