On Adaptive Bitprobe Schemes for Storing Two Elements

Abstract

In this paper, we look into the problem of storing a subset $\mathcal {S}$ containing at most two elements of the universe $\mathcal {U}$ in the adaptive bitprobe model. Due to the work of Radhakrishnan et al. [3], and more recently of Lewenstein et al. [2], we have excellent schemes for storing such an $\mathcal {S}$, and answering membership queries using two or more bitprobes. Yet, Nicholson et al. [4] in their survey of the area noted that the space lower bound of even the first non-trivial scenario, namely that of answering membership of $\mathcal {S}$ using two bitprobes, is still open. Towards that end, we propose an unified geometric approach to designing schemes in this domain. If t is the number of bitprobes allowed, we arrange the universe $\mathcal {U}$ in a $(2t-1)$-dimensional hypercube, and look at its two-dimensional faces. This approach matches the space bound of the best known schemes for certain cases, and gives results that are close to the best known schemes for others.