Combinatorial Group Testing and Its Applications

Combinatorial Group Testing and Its Applications / Edition 2

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General algorithms ch. Algorithms for special cases ch. Nonadaptive algorithms and binary superimposed codes ch. Multiaccess channels and extensions ch. Some other group testing models ch.

Combinatorial Group Testing and Its Applications - Dingzhu Du, Frank Hwang - Google Books

Competitive group testing ch. Unreliable tests ch. Optimal search in one variable ch. Unbounded search ch. Group testing on graphs ch. Membership problems ch. Complexity issues. Summary Group testing was first proposed for blood tests, but soon found its way to many industrial applications.

Combinatorial group testing studies the combinatorial aspect of the problem and is particularly related to many topics in combinatorics, computer science and operations research. Recently, the idea of combinatorial group testing has been applied to experimental designs, coding, multiaccess computer communication, done library screening and other fields. This book attempts to cover the theory and applications of combinatorial group testing in one place.

On the other hand, if no one in the pool has syphilis then many tests are saved, since every soldier in that group can be eliminated with just one test. The items that cause a group to test positive are generally called defective items these are the broken lightbulbs, syphilitic men, etc. There are two independent classifications for group-testing problems; every group-testing problem is either adaptive or non-adaptive, and either probabilistic or combinatorial.

In probabilistic models, the defective items are assumed to follow some probability distribution and the aim is to minimise the expected number of tests needed to identify the defectiveness of every item. On the other hand, with combinatorial group testing, the goal is to minimise the number of tests needed in a 'worst-case scenario' — that is, create a minmax algorithm — and no knowledge of the distribution of defectives is assumed.

The other classification, adaptivity, concerns what information can be used when choosing which items to group into a test. In general, the choice of which items to test can depend on the results of previous tests, as in the above lightbulb problem. An algorithm that proceeds by performing a test, and then using the result and all past results to decide which next test to perform, is called adaptive.

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Conversely, in non-adaptive algorithms, all tests are decided in advance. This idea can be generalised to multistage algorithms, where tests are divided into stages, and every test in the next stage must be decided in advance, with only the knowledge of the results of tests in previous stages. Although adaptive algorithms offer much more freedom in design, it is known that adaptive group-testing algorithms do not improve upon non-adaptive ones by more than a constant factor in the number of tests required to identify the set of defective items.

There are many ways to extend the problem of group testing. One of the most important is called noisy group testing, and deals with a big assumption of the original problem: that testing is error-free. A group-testing problem is called noisy when there is some chance that the result of a group test is erroneous e. Group testing can be extended by considering scenarios in which there are more than two possible outcomes of a test. Another extension is to consider geometric restrictions on which sets can be tested. The above lightbulb problem is an example of this kind of restriction: only bulbs that appear consecutively can be tested.

Similarly, the items may be arranged in a circle, or in general, a net, where the tests are available paths on the graph. Another kind of geometric restriction would be on the maximum number of items that can be tested in a group, [a] or the group sizes might have to be even and so on. In a similar way, it may be useful to consider the restriction that any given item can only appear in a certain number of tests.

Combinatorial Group Testing and Its Applications

There are endless ways to continue remixing the basic formula of group testing. The following elaborations will give an idea of some of the more exotic variants. In the 'good—mediocre—bad' model, each item is one of 'good', 'mediocre' or 'bad', and the result of a test is the type of the 'worst' item in the group. In threshold group testing, the result of a test is positive if the number of defective items in the group is greater than some threshold value or proportion.

Here, there is a third class of items called inhibitors, and the result of a test is positive if it contains at least one defective and no inhibitors. The concept of group testing was first introduced by Robert Dorfman in in a short report [2] published in the Notes section of Annals of Mathematical Statistics.

The method was simple: put the soldiers into groups of a given size, and use individual testing testing items in groups of size one on the positive groups to find which were infected. Dorfman tabulated the optimum group sizes for this strategy against the prevalence rate of defectiveness in the population. After , group testing remained largely untouched for a number of years. Then, the remaining items in the group are tested together, since it is very likely that none of them are defective. The first thorough treatment of group testing was given by Sobel and Groll in their formative paper on the subject.

The paper also made the connection between group testing and information theory for the first time, as well as discussing several generalisations of the group-testing problem and providing some new applications of the theory. The idea was to remove all the items in negative tests, and divide the remaining items into groups as was done with the initial pool. Combinatorial group testing in general was later studied more fully by Katona in In general, finding optimal algorithms for adaptive combinatorial group testing is difficult, and although the computational complexity of group testing has not been determined, it is suspected to be hard in some complexity class.

As such, the problem of adaptive combinatorial group testing — with a known number or upper bound on the number of defectives — has essentially been solved, with little room for further improvement. There is an open question as to when individual testing is minmax. One of the key insights in non-adaptive group testing is that significant gains can be made by eliminating the requirement that the group-testing procedure be certain to succeed the "combinatorial" problem , but rather permit it to have some low but non-zero probability of mis-labelling each item the "probabilistic" problem.

It is known that as the number of defective items approaches the total number of items, exact combinatorial solutions require significantly more tests than probabilistic solutions — even probabilistic solutions permitting only an asymptotically small probability of error. In this vein, Chan et al. Chan et al. Aldridge, Baldassini and Johnson produced an extension of the COMP algorithm that added additional post-processing steps. This leads on to the next definition. This is called the information lower bound. However, the information lower bound itself is usually unachievable, even for small problems.

In fact, the information lower bound can be generalised to the case where there is a non-zero probability that the algorithm makes an error. In this form, the theorem gives us an upper bound on the probability of success based on the number of tests. Algorithms for non-adaptive group testing consist of two distinct phases. First, it is decided how many tests to perform and which items to include in each test. In the second phase, often called the decoding step, the results of each group test are analysed to determine which items are likely to be defective.

The first phase is usually encoded in a matrix as follows. The matrix representation makes it possible to prove some bounds on non-adaptive group testing.

However, it does not guarantee that this will be straightforward. This means there is a simple procedure for finding the defectives: just remove every item that appears in a negative test. In fact, the generalised binary-splitting algorithm is close to optimal in the following sense. Non-adaptive group-testing algorithms tend to assume that the number of defectives, or at least a good upper bound on them, is known.

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Hwang: Combinatorial Group Testing and its Applications, 2nd. New combinatorial structures with applications to efficient group testing with inhibitors.

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Frequently used in a wide range of embedded applications requiring high levels. Time, the accumulated correctness behavior along with their resource. The set F not only influences the models accuracy, its computational cost, but also. Group testing is a combinatorial search paradigm 7 in which one wants to. NAGT is a natural paradigm and has found numerous applications in many. As HIV, it is only one of the many applications found for group testing: In the effort of. Recently, many applications of group testing have been found in molecular biol.

For such applications, it is often the case that a few abnormal traffic flows with. The counterpart to PGT is the combinatorial group testing. CGT formulated. Robert Dorfmans paper in introduced the field of Combinatorial. K, Combinatorial group testing and its applications, volume 12 of Series. Group testing has been used in medical, chemical and electrical testing, coding, drug screening, pollution control, multiaccess channel management, and. Combinatorial group testing CGT and its connection to the non-adaptive.

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