A Strong Direct Product Theorem for Disjointness [article]

Hartmut Klauck
2010 arXiv   pre-print
A strong direct product theorem states that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const· kn the success probability of solving k instances of size n can only be exponentially small in k. We show that this bound even
more » ... olds for AM communication protocols with limited ambiguity. This also implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a so-called Intersection Sampling Lemma that generalizes a result by Razborov.
arXiv:0908.2940v3 fatcat:peg2x6wjjffgfcuswgihxyvlsu