A Framework for Building Data Structures from Communication Protocols

We present a general framework for designing efficient data structures for high-dimensional pattern-matching problems (λ i ϵ [n], f(x_i,y) = 1) through communication models in which f(x,y) admits sublinear communication protocols with exponentially-small error. Specifically, we reduce the data structure problem to the Unambiguous Arthur-Merlin (UAM) communication complexity of f(x,y) under product distributions. We apply our framework to the Partial Match problem (a.k.a, matching with wildcards), whose underlying communication problem is sparse set-disjointness. When the database consists of n points in dimension d, and the number of ∗'s in the query is at most w = c logn λ€.,(λ‰ d), the fastest known linear-space data structure (Cole, Gottlieb and Lewenstein, STOC'04) had query time t ≈ 2^w = n^c, which is nontrivial only when c<1. By contrast, our framework produces a data structure with query time n^{1-1/(c log2 c)} and space close to linear. To achieve this, we develop a one-sided ϵ-error communication protocol for set-disjointness under product distributions with (λd log(1/ϵ)) complexity, improving on the classical result of Babai, Frankl and Simon (FOCS'86). Building on this protocol, we show that the Unambiguous AM communication complexity of w-Sparse set-disjointness with ϵ-error under product distributions is Õ(λw log(1/ϵ)), independent of the ambient dimension d, which is crucial for the partial match result. Our framework sheds further light on the power of data-dependent data structures, which is instrumental for reducing to the (much easier) case of product distributions.