Query-to-Communication Lifting Using Low-Discrepancy Gadgets

Abstract: Lifting theorems are theorems that relate the query complexity of a function $f:{0,1}^{n}\to{0,1}$ to the communication complexity of the composed function $f \circ g^{n}$, for some "gadget" $g:{0,1}^{b}\times{0,1}^{b}\to{0,1}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.