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تسجيل مستخدم جديدEfficient (nonrandom) construction and decoding for non-adaptive group testing
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The task of non-adaptive group testing is to identify up to $d$ defective items from $N$ items, where a test is positive if it contains at least one defective item, and negative otherwise. If there are $t$ tests, they can be represented as a $t times N$ measurement matrix. We have answered the question of whether there exists a scheme such that a larger measurement matrix, built from a given $ttimes N$ measurement matrix, can be used to identify up to $d$ defective items in time $O(t log_2{N})$. In the meantime, a $t times N$ nonrandom measurement matrix with $t = O left(frac{d^2 log_2^2{N}}{(log_2(dlog_2{N}) - log_2{log_2(dlog_2{N})})^2} right)$ can be obtained to identify up to $d$ defective items in time $mathrm{poly}(t)$. This is much better than the best well-known bound, $t = O left( d^2 log_2^2{N} right)$. For the special case $d = 2$, there exists an efficient nonrandom construction in which at most two defective items can be identified in time $4log_2^2{N}$ using $t = 4log_2^2{N}$ tests. Numerical results show that our proposed scheme is more practical than existing ones, and experimental results confirm our theoretical analysis. In particular, up to $2^{7} = 128$ defective items can be identified in less than $16$s even for $N = 2^{100}$.
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