Efficient Deterministic Quantitative Group Testing for Precise Information Retrieval

Abstract: The Quantitative Group Testing (QGT) is about learning a (hidden) subset $K$ of some large domain $N$ using a sequence of queries, where a result of a query provides information about the size of the intersection of the query with the unknown subset $K$. Almost all previous work focused on randomized algorithms minimizing the number of queries; however, in case of large domains $N$, randomization may result in a significant deviation from the expected precision. Others assumed unlimited computational power (existential results) or adaptiveness of queries. In this work we propose efficient non-adaptive deterministic QGT algorithms for constructing queries and deconstructing a hidden set $K$ from the results of the queries, without using randomization, adaptiveness or unlimited computational power. The efficiency is three-fold. First, in terms of almost-optimal number of queries - we improve it by factor nearly $|K|$ comparing to previous constructive results. Second, our algorithms construct the queries and reconstruct set $K$ in polynomial time. Third, they work for any hidden set $K$, as well as multi-sets, and even if the results of the queries are capped at $\sqrt{|K|}$. We also analyze how often elements occur in queries and its impact to parallelization and fault-tolerance of the query system.