This paper considers the problem of Quantitative Group Testing (QGT). Consider a set of N items among which K items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. In this work, we propose a nonadaptive QGT algorithm using sparse graph codes over biregular bipartite graphs with left-degree ℓ and right degree r and binary t-error-correcting BCH codes. The proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, we show that for the sub-linear regime where K N vanishes as K, N → ∞, the proposed algorithm requires at most m = c(t)K t log 2 ℓN c(t)K + 1 + 1 + 1 tests to recover all the defective items with probability approaching one as K, N → ∞, where c(t) depends only on t. The results of our theoretical analysis reveal that the minimum number of required tests is achieved by t = 2. The encoding and decoding of the proposed algorithm for any t ≤ 4 have the computational complexity of O(K log 2 N K ) and O(K log N K ), respectively. Our simulation results also show that the proposed algorithm significantly outperforms a non-adaptive semi-quantitative group testing algorithm recently proposed by Abdalla et al. in terms of the required number of tests for identifying all the defective items with high probability.

This paper considers the problem of Quantitative Group Testing (QGT) where there are some defective items among a large population of N items. We consider the scenario in which each item is defective with probability K/N , independently from the other items. In the QGT problem, the goal is to identify all or a sufficiently large fraction of the defective items by testing groups of items, with the minimum possible number of tests. In particular, the outcome of each test is a non-negative integer which indicates the number of defective items in the tested group. In this work, we propose a nonadaptive QGT scheme for the underlying randomized model for defective items, which utilizes sparse graph codes over irregular bipartite graphs with optimized degree profiles on the left nodes of the graph as well as binary t-error-correcting BCH codes. We show that in the sub-linear regime, i.e., when the ratio K/N vanishes as N grows unbounded, the proposed scheme with m = c(t, d)K(t log( ℓN c(t,d)K + 1) + 1) tests can identify all the defective items with probability approaching 1, where d and ℓ are the maximum and average left degree, respectively, and c(t, d) depends only on t and d (and does not depend on K and N ). For any t ≤ 4, the testing and recovery algorithms of the proposed scheme have the computational complexity of O(N log N K ) and O(K log N K ), respectively. The proposed scheme outperforms two recently proposed non-adaptive QGT schemes for the sub-linear regime, including our scheme based on regular bipartite graphs and the scheme of Gebhard et al., in terms of the number of tests required to identify all defective items with high probability.

In many practical settings, the user needs to retrieve information from a server in a periodic manner, over multiple rounds of communication. In this paper, we discuss the setting in which this information needs to be retrieved privately, such that the identity of all the information retrieved until the current round is protected. This setting can occur in practical situations in which the user needs to retrieve items from the server or a periodic basis, such that the privacy needs to be guaranteed for all the items been retrieved until the current round. We refer to this setting as an online private information retrieval as the user does not know the identities of the future items that need to be retrieved from the server.Following the previous line of work by Kadhe et al. we assume that the user knows a random subset of M messages in the database as a side information which are unknown to the server. Focusing on scalar-linear settings, we characterize the per-round capacity, i.e., the maximum achievable download rate at each round, and present a coding scheme that achieves this capacity. The key idea of our scheme is to utilize the data downloaded during the current round as a side information for the subsequent rounds. We show for the setting with K messages stored at the server, the per-round capacity of the scalar-linear setting is C1 = (M + 1)/K for round i = 1 and Ci = (2 i−1 (M + 1))/KM for round i ≥ 2, provided that K/(M + 1) is a power of 2.

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