Efficiently decodable non-adaptive group testing

We consider the following "efficiently decodable" non-adaptive group testing problem. There is an unknown string χ ∈ {0, 1} ⁿ with at most d ones in it. We are allowed to test any subset S ⊆ [n] of the indices. The answer to the test tells whether χ_i = 0 for all i ∈ S or not. The objective is to design as few tests as possible (say, t tests) such that χ can be identified as fast as possible (say, poly(t)-time). Efficiently decodable non-adaptive group testing has applications in many areas, including data stream algorithms and data forensics. A non-adaptive group testing strategy can be represented by a 1 x n matrix, which is the stacking of all the characteristic vectors of the tests. It is well-known that if this matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input string. Furthermore, we know how to construct d-disjunct matrices with t = O(d² log n) efficiently. However, these matrices so far only allow for a "decoding" time of O(nt), which can be exponentially larger than poly(t) for relatively small values of d. This paper presents a randomness efficient construction of d-disjunct matrices with t = O(d² log n) that can be decoded in time poly(d)·t log² t + O(t²). To the best of our knowledge, this is the first result that achieves an efficient decoding time and matches the best known O(d² log n) bound on the number of tests. We also derandomize the construction, which results in a polynomial time deterministic construction of such matrices when d = O(log n/ log log n). A crucial building block in our construction is the notion of (d, ℓ)-list disjunct matrices, which represent the more general "list group testing" problem whose goal is to output less than d + ℓ positions in χ, including all the (at most d) positions that have a one in them. List disjunct matrices turn out to be interesting objects in their own right and were also considered independently by [Cheraghchi, FCT 2009]. We present connections between list disjunct matrices, expanders, dispersers and disjunct matrices. List disjunct matrices have applications in constructing (d, ℓ)-sparsity separator structures [Ganguly, ISAAC 2008] and in constructing tolerant testers for Reed-Solomon codes in the data stream model.