Improved Explicit Data Structures in the Bitprobe Model

Buhrman et al. [SICOMP 2002] studied the membership problem in the bitprobe model, presenting both randomized and deterministic schemes for storing a set of size n from a universe of size m such that membership queries on the set can be answered using t bit probes. Since then, there have been several papers focusing on deterministic schemes, especially for the first non-trivial case when n = 2. The most recent, due to Radhakrishnan, Shah, and Shannigrahi [ESA 2010], describes non-explicit schemes (existential results) for t ≥ 3 using probabilistic arguments. We describe a fully explicit scheme for n = 2 that matches their space bound of Θ(m 2/5) bits for t = 3 and, furthermore, improves upon it for t > 3, answering their open problem. Our structure (consisting of query and storage algorithms) manipulates blocks of bits of the query element in a novel way that may be of independent interest. We also describe recursive schemes for n ≥ 3 that improve upon all previous fully explicit schemes for a wide range of parameters.