Group Testing With Random Pools: Optimal Two-Stage Algorithms

We study the group testing of a set of N items each of which is defective with probability p. We focus on the double limit of small defect probability, p ≪ 1, and large number of variables, N ≫ 1, taking either p → 0 after N → ∝ or p = 1/Nβ with β ∈ (0,1/2). In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, T̅(N, p), is known to scale as Np |log p|. Here we determine the sharp asymptotic value of T̅(N,p)/(Np|log p|) and construct a class of two-stage algorithms over which this optimal value is attained. This is done by choosing a proper bipartite regular graph (of tests and variable nodes) for the first stage of the detection. Furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree and the tests connected to a given variable are randomly chosen with uniform distribution among all tests. Finally, we improve the existing upper and lower bounds for the optimal number of tests in the case p = 1/Nβ with β ∈ [1/2,1).

[1]  R. Dorfman.  The Detection of Defective Members of Large Populations , 1943 .

[2]  M. Sobel,et al.  Group testing to eliminate efficiently all defectives in a binomial sample , 1959 .

[3]  Richard C. Singleton,et al.  Nonrandom binary superimposed codes , 1964, IEEE Trans. Inf. Theory.

[4]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[5]  H. Poincaré,et al.  Percolation ? , 1982 .

[6]  Jack K. Wolf,et al.  Born again group testing: Multiaccess communications , 1985, IEEE Trans. Inf. Theory.

[7]  E. Barillot,et al.  Theoretical analysis of library screening using a N-dimensional pooling strategy. , 1991, Nucleic acids research.

[8]  D. Du,et al.  Combinatorial Group Testing and Its Applications , 1993 .

[9]  Pavel A. Pevzner,et al.  Towards DNA Sequencing Chips , 1994, MFCS.

[10]  Emanuel Knill,et al.  Lower bounds for identifying subset members with subset queries , 1994, SODA '95.

[11]  Steven Skiena,et al.  Reconstructing strings from substrings in rounds , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[12]  D. Balding,et al.  Efficient pooling designs for library screening. , 1994, Genomics.

[13]  Emanuel Knill,et al.  A Comparative Survey of Non-Adaptive Pooling Designs , 1996 .

[14]  L. Wein,et al.  Pooled testing for HIV prevalence estimation: exploiting the dilution effect. , 2015, Statistics in medicine.

[15]  Richard E. Ladner,et al.  Group testing for image compression , 2000, Proceedings DCC 2000. Data Compression Conference.

[16]  Toby Berger,et al.  Asymptotic efficiency of two-stage disjunctive testing , 2002, IEEE Trans. Inf. Theory.

[17]  Toby Berger,et al.  Application of Cover-free Codes and Combinatorial Designs to Two-stage Testing , 2003, Discret. Appl. Math..

[18]  Vladimir I. Levenshtein,et al.  A universal bound for a covering in regular posets and its application to pool testing , 2003, Discret. Math..

[19]  Anna Scaglione,et al.  On multiple access for distributed dependent sources: a content-based group testing approach , 2004, Information Theory Workshop.

[20]  Annalisa De Bonis,et al.  Optimal Two-Stage Algorithms for Group Testing Problems , 2005, SIAM J. Comput..

[21]  J.M.F. Moura,et al.  Structured LDPC codes for high-density recording: large girth and low error floor , 2006, IEEE Transactions on Magnetics.

[22]  W. Marsden.  I and J , 2012 .