Multilevel group testing via sparse-graph codes

Abstract

In this paper, we consider the problem of multi-level group testing, where the goal is to recover a set of K defective items in a set of n items by pooling groups of items and observing the result of each test. The main difference of multilevel group testing with the classical non-adaptive group testing problem is that the result of each test is an integer in the set [L] = {0,1, • • •, L}: if there are i < L defective items in the pool, the result of the test is i, and if there are more than L items in the pool, the result of the test is L. We develop a multilevel group testing algorithm using sparse-graph codes that has low sample and computational complexity. More precisely, with high probability, our algorithm provably recovers (1 — ∊) fraction of the defective items using C(∊, L)K log(n) tests, where C(∊, L) is a constant that only depends on ∊ and the number of levels L, and it can be precisely characterized for arbitrary L and e. Furthermore, the computational complexity of our algorithm is O(K log(n)). As an example, our algorithm is able to recover (1 — 10−3) fraction of the defective items with only 13.8K log(n) measurements for L = 2. We also provide numerical results that show tight agreement with our theoretical results.