Efficient (nonrandom) Construction and Decoding for Non-adaptive Group Testing

抄録

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 × N measurement matrix. We have answered the question of whether there exists a scheme such that a larger measurement matrix, built from a given t × N measurement matrix, can be used to identify up to d defective items in time O(t log₂ N). In the meantime, a t × N nonrandom measurement matrix with $t = O \left(\frac{d^2 \log_2^2{N}}{(\log_2(d\log_2{N}) - \log_2{\log_2(d\log_2{N})})^2} \right)$ can be obtained to identify up to d defective items in time 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 $4\log_2^2{N}$ using $t = 4\log_2^2{N}$ 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 2⁷ = 128 defective items can be identified in less than 16s even for N = 2¹⁰⁰.