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Optimal Binomial Group Testing with a Test History

Published online by Cambridge University Press:  27 July 2009

B. Chen
Affiliation:
Institute of Applied Mathematics, Beijing, China
X. D. Hu
Affiliation:
Academia Sinica Beijing, China
F. K. Hwang
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey 07974

Abstract

The classical binomial group-testing problem considers the use of a minimal number of group tests to identify all defectives in a set of items, each of which independently has probability p of being defective. The minimization problem is very difficult, and the only major result is that for p ≥ (3 - □ root;5) /2, Unger proved that one-by-one testing is the best. Recently, to study the complexity of the group-testing problem, researchers have considered the more general case in which the items have already been tested. Interestingly, Unger's result still holds in this case. However, since the items are no longer equivalent (nor independent), the sequencing of items of testing becomes a legitimate problem, one that did not arise in the classical case. In this paper, we give optimal sequencing under general conditions on the test history.

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Articles
Copyright
Copyright © Cambridge University Press 1990

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References

Dorfman, R. (1943). The detection of defective members of large population. Annals of Mathematical Statistics 14: 436440.CrossRefGoogle Scholar
Du, D.Z. & Ko, K.I. (1987). Some completeness results on decision trees and group testing. SIAM Journal of Algebraic and Discrete Methods 8: 762777.CrossRefGoogle Scholar
Sobel, M. & Groll, P.A. (1959). Group testing to eliminate efficiently all defectives in a binomial sample. Bell System Technical Journal 35: 11791252.CrossRefGoogle Scholar
Unger, P. (1960). The cutoff point for group testing. Communication on Pure and Applied Mathematics 13: 4954.CrossRefGoogle Scholar
Yao, Y.C. & Hwang, F.K. (1988). Individual testing of independent items in optimal group testing. Probabilities in the Engineering and Informational Sciences 2: 2329.CrossRefGoogle Scholar

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