Non-Adaptive Randomized Algorithm for Group Testing
This work addresses the challenge of efficient decoding in group testing, which is incremental but offers practical gains for applications like disease screening or error detection.
The paper tackles the problem of group testing by introducing a new property called semi-disjunction for non-adaptive randomized algorithms in the random incidence design model, achieving linear-time decoding and optimal test numbers as d approaches infinity, with improvements over existing methods even for small d.
We study the problem of group testing with a non-adaptive randomized algorithm in the random incidence design (RID) model where each entry in the test is chosen randomly independently from $\{0,1\}$ with a fixed probability $p$. The property that is sufficient and necessary for a unique decoding is the separability of the tests, but unfortunately no linear time algorithm is known for such tests. In order to achieve linear-time decodable tests, the algorithms in the literature use the disjunction property that gives almost optimal number of tests. We define a new property for the tests which we call semi-disjunction property. We show that there is a linear time decoding for such test and for $d\to \infty$ the number of tests converges to the number of tests with the separability property and is therefore optimal (in the RID model). Our analysis shows that, in the RID model, the number of tests in our algorithm is better than the one with the disjunction property even for small $d$.