Abstract
We study the group testing problem with non-adaptive randomized algorithms. Several models have been discussed in the literature to determine how to randomly choose the tests. For a model \(\mathcal{M}\), let \(m_\mathcal{M}(n,d)\) be the minimum number of tests required to detect at most d defectives within n items, with success probability at least \(1-\delta \), for some constant \(\delta \). In this paper, we study the measures
In the literature, the analyses of such models only give upper bounds for \(c_\mathcal{M}(d)\) and \(c_\mathcal{M}\), and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for \(c_\mathcal{M}(d)\) and \(c_\mathcal{M}\) for all the known modelsĀ \(\mathcal{M}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Rykov, V.V., Dāyachkov, A.G.: Bounds on the length of disjunctive codes. Probl. Peredachi Inf. 18, 7ā13 (1982)
Angluin, D.: Queries and concept learning. Mach. Learn. 2(4), 319ā342 (1987)
Balding, D.J., Bruno, W.J., Torney, D.C., Knill, E.: A comparative survey of non-adaptive pooling designs. In: Speed, T., Waterman, M.S. (eds.) Genetic Mapping and DNA Sequencing. IMA, vol. 81, pp. 133ā154. Springer, New York (1996). https://doi.org/10.1007/978-1-4612-0751-1_8
Bruno, W.J., et al.: Efficient pooling designs for library screening. Genomics 26(1), 21ā30 (1995)
Bshouty, N.H., Diab, N., Kawar, S.R., Shahla, R.J.: Non-adaptive randomized algorithm for group testing. In International Conference on Algorithmic Learning Theory, ALT 2017, 15ā17 October 2017, Kyoto University, Kyoto, Japan, pp. 109ā128 (2017)
Cicalese, F.: Group testing. Fault-Tolerant Search Algorithms. MTCSAES, pp. 139ā173. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-17327-1_7
Cormode, G., Muthukrishnan, S.: Whatās hot and whatās not: tracking most frequent items dynamically. ACM Trans. Database Syst. 30(1), 249ā278 (2005)
Damaschke, P., Muhammad, A.S.: Randomized group testing both query-optimal and minimal adaptive. In: BielikovĆ”, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., TurĆ”n, G. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 214ā225. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-27660-6_18
Dorfman, R.: The detection of defective members of large populations. Ann. Math. Stat. 14(4), 436ā440 (1943)
Du, D.-Z., Hwang, F.K.: Combinatorial Group Testing and Its Applications. World Scientfic Publishing, Singapore (1993)
Du, D.-Z., Hwang, F.K.: Pooling Designs and Nonadaptive Group Testing: Important Tools for DNA Sequencing. World Scientfic Publishing, Singapore (2006)
Erdƶs, P., RĆ©nyi, A.: On two problems of information theory, pp. 241ā254 (1963). Publications of the Mathematical Institute of the Hungarian Academy of Sciences
FĆ¼redi, Z.: On \(r\)-cover-free families. J. Comb. Theory Ser. A 73(1), 172ā173 (1996)
Hong, E.S., Ladner, R.E.: Group testing for image compression. IEEE Trans. Image Process. 11(8), 901ā911 (2002)
Hwang, F.K.: A method for detecting all defective members in a population by group testing. J. Am. Stat. Assoc. 67(339), 605ā608 (1972)
Hwang, F.K.: Random k-set pool designs with distinct columns. Probab. Eng. Inf. Sci. 14(1), 49ā56 (2000)
Hwang, F.K., Liu, Y.C.: The expected numbers of unresolved positive clones for various random pool designs. Probab. Eng. Inf. Sci. 15(1), 57ā68 (2001)
Hwang, F.K., Liu, Y.C.: A general approach to compute the probabilities of unresolved clones in random pooling designs. Probab. Eng. Inf. Sci. 18(2), 161ā183 (2004)
Hwang, F.K., Liu, Y.: Random pooling designs under various structures. J. Comb. Optim. 7, 339ā352 (2003)
Kautz, W., Singleton, R.: Nonrandom binary superimposed codes. IEEE Trans. Inf. Theory 10(4), 363ā377 (1964)
Macula, A.J., Popyack, L.J.: A group testing method for finding patterns in data. Discrete Appl. Math. 144(1), 149ā157 (2004). Discrete Mathematics and Data Mining
Porat, E., Rothschild, A.: Explicit nonadaptive combinatorial group testing schemes. IEEE Trans. Inf. Theory 57(12), 7982ā7989 (2011)
Ngo, H.Q., Du, D.-Z.: A survey on combinatorial group testing algorithms with applications to DNA library screening. Discrete Math. Theor. Comput. Sci. 55, 171ā182 (2000). DIMACS Series
RuszinkĆ³, M.: On the upper bound of the size of the r-cover-free families. J. Comb. Theory Ser. A 66(2), 302ā310 (1994)
Sebƶ, A.: On two random search problems. J. Stat. Plan. Inference 11(1), 23ā31 (1985)
Wolf, J.: Born again group testing: multiaccess communications. IEEE Trans. Inf. Theory 31(2), 185ā191 (1985)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Bshouty, N.H., Haddad, G., Haddad-Zaknoon, C.A. (2020). Bounds for the Number of Tests in Non-adaptive Randomized Algorithms for Group Testing. In: Chatzigeorgiou, A., et al. SOFSEM 2020: Theory and Practice of Computer Science. SOFSEM 2020. Lecture Notes in Computer Science(), vol 12011. Springer, Cham. https://doi.org/10.1007/978-3-030-38919-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-38919-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38918-5
Online ISBN: 978-3-030-38919-2
eBook Packages: Computer ScienceComputer Science (R0)