Optimal Two-Stage Algorithms for Group Testing Problems

Abstract

Group testing refers to the situation in which one is given a set of objects O, an unknown subset P ⊆ O, and the task of determining P by asking queries of the type “does P intersect Q?”, where Q is a subset of O. Group testing is a basic search paradigm that occurs in a variety of situations such as quality control testing, searching in storage systems, multiple access communications, and data compression, among others. Group testing procedures have been recently applied in computational molecular biology, where they are used for screening libraries of clones with hybridization probes and sequencing by hybridization. Motivated by particular features of group testing algorithms used in biological screening, we study the efficiency of two-stage group testing procedures. Our main result is the first optimal two- stage algorithm that uses a number of tests of the same order as the information-theoretic lower bound on the problem. We also provide efficient algorithms for the case in which there is a Bernoulli probability distribution on the possible sets P, and an optimal algorithm for the case in which the outcome of tests may be unreliable because of the presence of “inhibitory” items in O. Our results depend on a combinatorial structure introduced in this paper. We believe that it will prove useful in other contexts, too

An extension of Stein-Lovász theorem and some of its applications

“…The Stein-Lovász theorem was used in dealing with the upper bounds for the sizes of (k, m, n)-selectors (De Bonis et al 2005). Inspired by this work, it was also used in dealing with the upper bounds for the sizes of (d, r; z]-disjunct matrices (Chen et al 2008).…”

“…The Stein-Lovász theorem was used in dealing with the upper bounds for the sizes of (k, m, n)-selectors (De Bonis et al 2005), followed by the upper bounds for the sizes of (d, r; z]-disjunct matrices (Chen et al 2008). More of its applications for treating the minimal sizes of various set systems were also found in Deng et al (2011).…”

“…These applications were obtained by translating the problems into the hypergraph problems and showing that the models can be deduced from the vertex cover of properly defined hypergraphs. The notion of (k, m, n)-selectors was first introduced by De Bonis et al (2005), followed by a generalization to the notion of (k, m, w, n)-selectors (De Bonis 2008). In this paper, we use the notation (k, m, c, n)-selectors for convenience.…”

Exact learning of juntas from membership queries

“…This problem was studied in [8,9,10,4]. In this paper, we find new bounds, tighten some of the gaps between some lower bounds and upper bounds 1 and find new algorithms with better query and time complexities both for the deterministic and the randomized case.…”

“…See many other applications in [2,3,4,5,6,7]. Many of the new applications raised new models and new problems and in many of those applications the function being learned can be an arbitrary function that depends on few variables.…”