Efficiently Decodable Non-Adaptive Croup Testing

We consider die following "efficiently decodable" non-adaptive group testing problem There is an unknown string x is an element of {0, 1}(n) with at most d ones in it We are allowed to test any subset S C [n] of the indices The answer to the test tells whetherr x(i) = 0 for all iota is an element of S or not The objective is 1,0 design as few tests as possible (say, t tests) such that a; can be identified as fast as possible (say, poly(l)-time) Efficiently. decodable non-adaptive group testing has applications in many areas. including data stream algorithms and data forensicsA non-adaptive group testing strategy can be represented by a t x n matrix, which is the stacking of all the characteristic vectors of the tests It is well-known that if tins matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input stung Put , we know how to construct (I-disjunct matrices with t = O(d(2) log n) efficiently However these matrices so fat only allow lot a. "decoding" time of O(nl), which can be exponentially larger than poly(t) for relatively small values of dThis impel presents a randomness efficient construction of d-disjunct matrices with t = O(d(2) log n) that, can be decoded in time poly(d) t log(2) t + O(t(2)) to the best of our knowledge, this is the first, result that. achieves an efficient decoding time and matches the best known 0(d2 log n) bound on the number of tests We also derandomize the construction, winch results in a polynomial time deterministic construction of such matrices when d = O(log n/log log n)A critical 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 positions in x, including all the (at most (I) positions that have a one in them. List disjunct matrices turn out, to be interesting objects in their own light and were also considered independently by [Cheraghchi. FCT 2009] We present connections between list disjunct matt ices, expanders, dispersers and disjunct matrices List disjunct matrices have applications in constructing (d, e)sparsity separator structures [Ganguly, ISAAC 2008] and in constructing tolerant testers or Reed-Solomon codes in the data stream model