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ASYMPTOTIC ANALYSIS OF OPTIMAL NESTED GROUP-TESTING PROCEDURES

Published online by Cambridge University Press:  29 June 2016

Nabil Zaman  and
Nicholas Pippenger
Nabil Zaman
Affiliation:
Harvey Mudd College, 301 Platt Blvd., Claremont, CA 91711, USA E-mail: nzaman@g.hmc.edu; njp@math.hmc.edu.
Nicholas Pippenger
Affiliation:
Harvey Mudd College, 301 Platt Blvd., Claremont, CA 91711, USA E-mail: nzaman@g.hmc.edu; njp@math.hmc.edu.
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Abstract

We analyze a construction for optimal nested group-testing procedures, and show that, when individuals are independently positive with probability p, the expected number of tests per positive individual, F*(p), has, as p→0, the asymptotic behavior

$$F^{\ast}(p) = \log_2 {1\over p} + \log_2 \log 2 + 2 + f\left(\log_2 {1\over p} + \log_2 \log 2\right) + O(p),$$
where
$$f(z) = 4\times 2^{-2^{1-\{z\}}} - \{z\} - 1,$$
 and {z}=z−⌊z⌋ is the fractional part of z. The function f(z) is a periodic function (with period 1) that exhibits small oscillations (with magnitude <0.005) about an even smaller average value (<0.0005).

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 4 , October 2016 , pp. 547 - 552
Copyright
Copyright © Cambridge University Press 2016 

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