Randomized group testing both query-optimal and minimal adaptive

The classical group testing problem asks to determine at most d defective elements in a set of n elements, by queries to subsets that return Yes if the subset contains some defective, and No if the subset is free of defectives. By the entropy lower bound, d log n tests are needed at least. We devise group testing strategies that combine two features: They achieve this optimal query bound asymptotically, with a factor 1+o(1) as n grows, and they work in a fixed number of stages of parallel queries. Our strategies are randomized and have a controlled failure probability, i.e., constant but arbitrarily small. We consider different settings (known or unknown d, probably correct or verified outcome), and we aim at the smallest possible number of stages. In particular, 2 stages are sufficient if d grows slowly enough with n, and 4 stages are sufficient if d=o(n)