Smart elements in combinatorial group testing problems

Abstract

In combinatorial group testing problems the questioner needs to find a special element x ∈ [n] by testing subsets of [n]. Tapolcai et al. [27,28] introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the special one.Using classical results of extremal set theory we prove that if F n ⊆ 2 [n] solves the nonadaptive version of this problem and has minimal cardinality, then lim n→∞ |F n | log 2 n = log (3/2) 2.This improves results in [27,28].We also consider related models inspired by secret sharing models, where the elements should share information among them to find out the special one. Finally the adaptive versions of the different models are investigated.and for F ⊆ [n] the answer is the appropriate value of the function t :The tested subsets are called queries and the special element is usually called defective in the group testing literature. Questioner's aim is to ask as few queries as possible and the number of queries needed to ask in the worst case is called the worst-case complexity of the problem. For any combinatorial group testing problem there are at least two main approaches: whether it is adaptive or non-adaptive. In the adaptive scenario Questioner asks queries depending on the answers for the previously asked queries, however in the non-adaptive version Questioner must pose all the queries at the beginning.Let us briefly describe the solution for the (above mentioned) most basic combinatorial group testing model in the non-adaptive case. We call a family F ⊆ 2 [n] separating if for any two different x, y ∈ [n] there is F ∈ F with x ∈ F and y ∈ F, or y ∈ F and x ∈ F.Fact 1. Questioner finds the defective by asking elements of F ⊆ 2 [n] if and only if F is separating.The notion of separating family in the context of combinatorial group testing was introduced and first studied by Rényi in [23]. We will also use the following simple fact later:Fact 2. Suppose F n ⊆ 2 [n] is the smallest separating family. Then we have:|F n | = ⌈log 2 n⌉.One can imagine many possible generalizations of the most basic classical model: more defectives, other answers (threshold [6] or density [13] group testing), average case complexity [4], rounds [7, 15]. For a survey on different non-adaptive models see e.g. [10]. Combinatorial group testing problems were first considered during the World War II by Dorfman [9] in the context of mass blood testing. Since then group testing techniques have had many different applications, for example in fault diagnosis in optical networks [17], in quality control in product testing [25] or failure detection in wireless sensor networks [21]. New feature of the elementsInspired by the node failure localization model of Tapolcai et al. [27,28] we introduce a possible new feature of the elements. Informally speaking an element can be kind of smart and this fact means two things:

Smart elements in combinatorial group testing problems with more defectives

“…Note that for d = 1 there is a solution for Model 2 d and there is no solution for Model 3 d [18]. For d ≥ 2 the situation is just the opposite: we will show that there is a solution for Model 3 d for n large enough.…”

“…In [18] the authors introduced new combinatorial group testing models, inspired by the results of Tapolcai et al [30,29].…”

“…Here we deal with a simple combinatorial version, where privacy only means that an unauthorized participant cannot completely detect the defective element(s). In [18] the authors considered models with one defective element. The main aim of this article is to continue these investigations with more defectives.…”

“…In this section we assume that there are exactly d ≥ 2 defective elements (and every element knows that). We consider models analogous to the ones introduced in [18].…”

“…• In [18] the authors considered the abstract version of the model introduced by Tapolcai et al [29,30]. Here we extended our models to the case of more defectives.…”