In group testing, the task is to determine the distinguished members of a set of objects O by asking subset queries of the form "does the set Q ` O contain a distinguished object?" In biological applications of group testing, the task is to repeatedly screen a library of objects for those which are positive for a probe. The subset queries consist of screening a pooled subset of the objects with the probe. This procedure has become an important component of the experimental methods used for the compilation of physical maps of chromosomes and other genetic material. For many screening applications, it is most cost-effective to ask many subset queries in parallel. This leads to non-adaptive group testing problems. An important aspect of most screening environments is that the screening results are far from reliable. In this report we discuss some of the error models that can be used and show how they affect the design of non-adaptive screening experiments. We give a unified treatment of ...

An instance of a group testing problem is a set of objects O and an unknown subset P of O. The task is to determine P by using queries of the type "does P intersect Q", where Q is a subset of O. This problem occurs in areas such as fault detection, multiaccess communications, optimal search, blood testing and chromosome mapping. Consider the two stage algorithm for solving a group testing problem. In the first stage a predetermined set of queries are asked in parallel and in the second stage, P is determined by testing individual objects. Let n = O . Suppose that P is generated by independently adding each x 2 O to P with probability p=n. Let q1 (q2) be the number of queries asked in the first (second) stage of this algorithm. We show that if q1 = o(log(n) log(n)= log log(n)), then Exp(q2) = n 1\Gammao(1) , while there exist algorithms with q1 = O(log(n) log(n)= log log(n)) and Exp(q2 ) = o(1). The proof involves a relaxation technique which can be used with arbitrary distributio...

Let X1,..., Xn be a collection of iid discrete random variables, and Y1,..., Ym a set of noisy observations of such variables. Assume each observation Ya to be a random function of some a random subset of the Xi’s, and consider the conditional distribution of Xi given the observations, namely µi(xi) ≡ P{Xi = xi|Y} (a posteriori probability). We establish a general decoupling principle among the Xi’s, as well as a relation between the distribution of µi, and the fixed points of the associated density evolution operator. These results hold asymptotically in the large system limit, provided the average number of variables an observation depends on is bounded. We discuss the relevance of our result to a number of applications, ranging from sparse graph codes, to multi-user detection, to group testing. 1

Abstract—We adapt methods originally developed in information and coding theory to solve some testing problems. The efficiency of two-stage pool testing of items is characterized by the minimum expected number @ A of tests for the Bernoulli-scheme, where the minimum is taken over a matrix that specifies the tests that constitute the first stage. An information-theoretic bound implies that the natural desire to achieve @ Aa @ A as can be satisfied only if @ A H. Using random selection and linear programming, we bound some parameters of binary matrices, thereby determining up to positive constants how the asymptotic behavior of @ A as depends on the manner in which @ A H. In particular, it is shown that for Aa C

Abstract — Detection of defective members of large populations has been widely studied in the statistics community under the name “group testing”, a problem which dates back to World War II when it was suggested for syphilis screening. There, the main interest is to identify a small number of infected people among a large population using collective samples. In viral epidemics, one way to acquire collective samples is by sending agents inside the population. While in classical group testing, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in this work we assume that the decoder possesses only partial knowledge about the sampling process. This assumption is justified by observing the fact that in a viral sickness, there is a chance that an agent remains healthy despite having contact with an infected person. Therefore, the reconstruction method has to cope with two different types of uncertainty; namely, identification of the infected population and the partially unknown sampling procedure. In this work, by using a natural probabilistic model for “viral infections”, we design non-adaptive sampling procedures that allow successful identification of the infected population with overwhelming probability 1 − o(1). We propose both probabilistic and explicit design procedures that require a “small ” number of agents to single out the infected individuals. More precisely, for a contamination probability p, the number of agents required by the probabilistic and explicit designs for identification of up to k infected members is bounded by m = O(k 2 (log n)/p 2) and m = O(k 2 (log 2 n)/p 2), respectively. In both cases, a simple decoder is able to successfully identify the infected population in time O(mn). I.

Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design non-adaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items.

Abstract — In this paper we consider the multiple access problem with distributed dependent sources. We derive the optimal designs for the case of N correlated binary sources whose data are modelled as a two-state Markov chain. The solution can be classified as a group testing technique where data values at the sensors are determined through the successive refinements of the tests over smaller groups. The tests form, progressively, an accurate map of the sensor data at the central receiver. We derive the conditions on the parameters of the data model for which the group testing approach is superior to time sharing. In contrast to standard multiple access techniques, this is the first method proposed for data retrieval from distributed dependent sources which is content-based rather than user-based. 1 I.

Abstract—Non-adaptive group testing involves grouping arbitrary subsets of n items into different pools and identifying defective items based on tests obtained for each pool. Motivated by applications in network tomography, sensor networks and infection propagation we formulate non-adaptive group testing problems on graphs. Unlike conventional group testing problems each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper we associate a test with a random walk. In this context conventional group testing corresponds to the special case of a complete graph on n vertices. For interesting classes of graphs we arrive at a rather surprising result, namely, that the number of tests required to identify d defective items is substantially similar to that required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if T (n) corresponds to the mixing time of the graph G, we show that with m = O(d 2 T 2 (n) log(n/d)) non-adaptive tests, one can identify the defective items. Consequently, for the Erdős-Rényi random graph G(n, p), as well as expander graphs with constant spectral gap, it follows that m = O(d 2 log 3 n) non-adaptive tests are sufficient to identify d defective items. We next consider a specific scenario that arises in network tomography and show that m = O(d 3 log 3 n) non-adaptive tests are sufficient to identify d defective items. We also consider noisy counterparts of the graph constrained group testing problem and develop parallel results for these cases. I.

Abstract We propose two new classes of non-adaptive pooling designs. The firstone is guaranteed to be d-error-detecting and thus b d2 c-error-correcting, where d, a positive integer, is the maximum number of defectives (or positives).Hence, the number of errors which can be detected grows linearly with the number of positives. Also, this construction induces a construction of a bi-nary code with minimum Hamming distance at least

We consider the following constrained version of the classical Group Testing Problem: Given a finite set of items identified with the set of natural numbers 2, . . . , n} and an unknown distinguished subset P 2, . . . , n} of up to p positive elements, the goal is to identify the items in P by asking the least number of queries of the type "does the subset Q 2, . . . , n} intersect P?", where Q is a subset of consecutive elements of 2, . . . , n} of cardinality at most d. This particular case of the Group Testing Problem naturally arises in several scenarios, most notably in Computational Biology. In this paper we focus on algorithms that solve the aforesaid problem and for which queries can be arranged in stages: in each stage a certain number of queries can be performed in parallel, while queries of a given stage can be chosen depending on the answers to those of previous stages. Algorithms that operate in few stages are usually preferred in practical applications. We study the case with one positive element comprehensively. For two-stage strategies for arbitrarily many positives, we obtain asymptotically tight bounds on the number of queries. Furthermore we prove upper bounds for any number of stages and positives, and we discuss the problem with the restriction that query intervals have some bounded length.