On the optimal pairwise group testing algorithm

Abstract

Originally suggested for the blood testing problem by Dorfman in 1943, an idea of Group Testing (GT) has found many applications in other fields as well. Among many (binomial) GT procedures introduced since then, in 1990, Yao and Hwang proposed the Pairwise Testing Algorithm (PTA) and demonstrated that PTA is the unique optimal nested GT procedure provided the probability of contamination lies in $\left(1-\frac{\sqrt{2}}{2},\frac{3-\sqrt{5}}{2}\right)$. Despite the fundamental nature of the result, PTA did not receive considerable attention in the literature. In particular, even its basic probabilistic properties remained unexplored. In this paper, we fill the gap. This is achieved through derivation of an explicit form of the moment generating function (MGF) for the total number of tests ${\mathit{T}}_{\mathit{n}}$ required by the PTA procedure. By making use of MGF, we obtain first moments of ${\mathit{T}}_{\mathit{n}}$ and characterize its asymptotic behaviour by establishing Law of Large Numbers, Central Limit Theorem and Large Deviation Principle.