Improved Algorithms for Group Testing with Inhibitors

Abstract Recent biological applications motivate a new group testing model where in addition to the category of the positive samples and the one of the negative samples, there is a third class of samples called inhibitors. The presence of positives in a test set cannot be detected if the test set contains one or more inhibitors. Let n be the total number of samples and p and r denote the number of positive and inhibitor samples, respectively. Farach et al. (1997), who introduced this model, have given a lower bound of Ω(log(( p n )( r n − p ))) on the number of tests required to find the p positives. They have also described a randomized algorithm to find the p positives which achieve this bound when p + r ⪡ n. In this paper, we give a better lower bound on the number of tests required to find the p positives by uncovering a relation between this group testing problem and cover-free families. We also provide efficient deterministic algorithms to find the positive samples.

[1]  Martin Aigner.  Combinatorial search , 1988 .

[2]  Ding-Zhu Du,et al.  Competitive Group Testing , 1993, Discret. Appl. Math..

[3]  Annalisa De Bonis,et al.  Group Testing with Unreliable Tests , 1997, Inf. Sci..

[4]  Zoltán Füredi,et al.  Families of Finite Sets in Which No Set Is Covered by the Union of Two Others , 1982, J. Comb. Theory, Ser. A.

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

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

[7]  Ding-Zhu Du,et al.  On Competitive Group Testing , 1994, SIAM J. Comput..

[8]  Peter Damaschke.  The Algorithmic Complexity of Chemical Threshold Testing , 1997, CIAC.

[9]  Frank K. Hwang,et al.  Comparing File Copies with at Most Three Disagreeing Pages , 1997, IEEE Trans. Computers.

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

[11]  Annalisa De Bonis,et al.  Improved Algorithms for Chemical Threshold Testing Problems , 1998, COCOON.

[12]  Chou Hsiung Li.  A Sequential Method for Screening Experimental Variables , 1962 .

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

[14]  Sundar Vishwanathan,et al.  Locality based graph coloring , 1993, STOC.

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

[16]  Alfredo De Santis,et al.  Visual Cryptography for General Access Structures , 1996, Inf. Comput..

[17]  Guoliang Xue,et al.  Modifications of Competitive Group Testing , 1994, SIAM J. Comput..

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

[19]  P. Erdös,et al.  Families of finite sets in which no set is covered by the union ofr others , 1985 .

[20]  Peter Damaschke.  A Parallel Algorithm for Nearly Optimal Edge Search , 1995, Inf. Process. Lett..

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

[22]  Amos Fiat,et al.  Tracing traitors , 2000, IEEE Trans. Inf. Theory.

[23]  Brian Gluss.  A Method for Obtaining Suboptimal Group-Testing Policies Using Dynamic Programming and Information Theory , 1963, JACM.

[24]  Andrew Chi-Chih Yao,et al.  An Almost Optimal Algorithm for Unbounded Searching , 1976, Inf. Process. Lett..

[25]  Miklós Ruszinkó,et al.  On the upper bound of the size of the r -cover-free families , 1994 .

[26]  Luisa Gargano,et al.  Search problems for two irregular coins with incomplete feedback: the underweight model , 1992, Discret. Appl. Math..

[27]  E. Sperner.  Ein Satz über Untermengen einer endlichen Menge , 1928 .