In statistics and combinatorial mathematics, group testing is any procedure that breaks up the task of identifying objects into tests on groups of items, rather than testing each item individually. First studied by Robert Dorfman in 1943, group testing is a relatively new field of mathematics that can be applied to a wide range of practical applications and is an active area of research today. A familiar example of group testing involves a string of light bulbs connected in series, where exactly one of the bulbs is known to be broken. The objective is to find the broken bulb using the smallest number of tests (where a test is when some of the bulbs are connected to a power supply). A simple approach is to test each bulb individually. However, when there are a large number of bulbs it would be much more efficient to pool the bulbs into groups. For example, by connecting the first half of the bulbs at once, it can be determined which half the broken bulb is in, ruling out half of the bulbs in just one test. Schemes for carrying out group testing can be simple or complex and the tests involved at each stage may be different. Schemes in which the tests for the next stage depend on the results of the previous stages are called adaptive procedures, while schemes designed so that all the tests are known beforehand are called non-adaptive procedures. The structure of the scheme of the tests involved in a non-adaptive procedure is known as a pooling design. Group testing has many applications, including statistics, biology, computer science, medicine, engineering and cyber security. Modern interest in these testing schemes has been rekindled by the Human Genome Project.
Basic description and terms Unlike many areas of mathematics, the origins of group testing can be traced back to a single report written by a single person: Robert Dorfman. The motivation arose during the Second World War when the United States Public Health Service and the Selective service embarked upon a large-scale project to weed out all syphilitic men called up for induction. Testing an individual for syphilis involves drawing a blood sample from them and then analysing the sample to determine the presence or absence of syphilis. At the time, performing this test was expensive, and testing every soldier individually would have been very expensive and inefficient. Supposing there are n {\displaystyle n} soldiers, this method of testing leads to n {\displaystyle n} separate tests. If a large proportion of the people are infected then this method would be reasonable. However, in the more likely case that only a very small proportion of the men are infected, a much more efficient testing scheme can be achieved. The feasibility of a more effective testing scheme hinges on the following property: the soldiers can be pooled into groups, and in each group the blood samples can be combined. The combined sample can then be tested to check if at least one soldier in the group has syphilis. This is the central idea behind group testing. If one or more of the soldiers in this group has syphilis, then a test is wasted (more tests need to be performed to find which soldier(s) it was). 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.). Often, the total number of items is denoted as n {\displaystyle n} and d {\displaystyle d} represents the number of defectives if it is assumed to be known.
Classification of group-testing problems 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. 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. In addition to this, non-adaptive methods are often useful in practice because one can proceed with successive tests without first analysing the results of all previous tests, allowing for the effective distribution of the testing process.
Variations and extensions 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.g. comes out positive when the test contained no defectives). The Bernoulli noise model assumes this probability is some constant, q {\displaystyle q} , but in general it can depend on the true number of defectives in the test and the number of items tested. For example, the effect of dilution can be modelled by saying a positive result is more likely when there are more defectives (or more defectives as a fraction of the number tested), present in the test. A noisy algorithm will always have a non-zero probability of making an error (that is, mislabeling an item). Group testing can be extended by considering scenarios in which there are more than two possible outcomes of a test. For example, a test may have the outcomes 0 , 1 {\displaystyle 0,1} and 2 + {\displaystyle 2^{+}} , corresponding to there being no defectives, a single defective, or an unknown number of defectives larger than one. More generally, it is possible to consider the outcome-set of a test to be 0 , 1 , … , k + {\displaystyle {0,1,\ldots ,k^{+}}} for some k ∈ N {\displaystyle k\in \mathbb {N} } . 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, 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. 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. Group testing with inhibitors is a variant with applications in molecular biology. 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.
History and development
Invention and initial progress The concept of group testing was first introduced by Robert Dorfman in 1943 in a short report published in the Notes section of Annals of Mathematical Statistics. Dorfman's report – as with all the early work on group testing – focused on the probabilistic problem, and aimed to use the novel idea of group testing to reduce the expected number of tests needed to weed out all syphilitic men in a given pool of soldiers. 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. Stephen Samuels found a closed-form solution for the optimal group size as a function of the prevalence rate. After 1943, group testing remained largely untouched for a number of years. Then in 1957, Sterrett produced an improvement on Dorfman's procedure. This newer process starts by again performing individual testing on the positive groups, but stopping as soon as a defective is identified. 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 1959 paper on the subject. They described five new procedures – in addition to generalisations for when the prevalence rate is unknown – and for the optimal one, they provided an explicit formula for the expected number of tests it would use. 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 fundamental result by Peter Ungar in 1960 shows that if the prevalence rate p > p u {\displaystyle p>p_{u}} , where p u = ( 3 − 5 ) / 2 ≈ 0.38 {\displaystyle p_{u}=(3-{\sqrt {5}})/2\approx 0.38} , then individual testing is the optimal group testing procedure with respect to the expected number of tests, and if p < p u {\displaystyle p<p_{u}} , then it is not optimal. However, it is important to note that despite 80 years' worth of research effort, the optimal procedure is yet unknown for p < p u {\displaystyle p<p_{u}} and a general population size n > 2 {\displaystyle n>2} .
Combinatorial group testing Group testing was first studied in the combinatorial context by Li in 1962, with the introduction of Li’s s {\displaystyle s} -stage algorithm. Li proposed an extension of Dorfman's '2-stage algorithm' to an arbitrary number of stages that required no more than t = e log 2 ( e ) d log 2 ( n ) {\textstyle t={\frac {e}{\log _{2}(e)}}d\log _{2}(n)} tests to be guaranteed to find d {\displaystyle d} or fewer defectives among n {\displaystyle n} items. 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. This was to be done s − 1 {\displaystyle s-1} times before performing individual testing. Combinatorial group testing in general was later studied more fully by Katona in 1973. Katona introduced the matrix representation of non-adaptive group-testing and produced a procedure for finding the defective in the non-adaptive 1-defective case in no more than t = ⌈ log 2 ( n ) ⌉ {\displaystyle t=\lceil \log _{2}(n)\rceil } tests, which he also proved to be optimal. 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. However, an important breakthrough occurred in 1972, with the introduction of the generalised binary-splitting algorithm. The generalised binary-splitting algorithm works by performing a binary search on groups that test positive, and is a simple algorithm that finds a single defective in no more than the information-lower-bound number of tests. In scenarios where there are two or more defectives, the generalised binary-splitting algorithm still produces near-optimal results, requiring at most d − 1 {\displaystyle d-1} tests above the information lower bound where d {\displaystyle d} is the number of defectives. Considerable improvements to this were made in 2013 by Allemann, getting the required number of tests to less than 0.187 d + 0.5 log 2 ( d ) + 5.5 {\displaystyle 0.187d+0.5\log _{2}(d)+5.5} above the information lower bound when n / d ≥ 38 {\displaystyle n/d\geq 38} and d ≥ 10 {\displaystyle d\geq 10} . This was achieved by changing the binary search in the binary-splitting algorithm to a complex set of sub-algorithms with overlapping test groups. 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. Hu, Hwang and Wang showed in 1981 that individual testing is minmax when n ≤ ⌊ ( 5 d + 1 ) / 2 ⌋ {\displaystyle n\leq \lfloor (5d+1)/2\rfloor } , and that it is not minmax when n > 3 d {\displaystyle n>3d} . It is currently conjectured that this bound is sharp: that is, individual testing is minmax if and only if n ≤ 3 d {\displaystyle n\leq 3d} . Some progress was made in 2000 by Riccio and Colbourn, who showed that for large n {\displaystyle n} , individual testing is minmax when d ≥ n / log 3 / 2 ( 3 ) ≈ 0.369 n {\displaystyle d\geq n/\log _{3/2}(3)\approx 0.369n} .
Non-adaptive and probabilistic testing 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. (2011) introduced COMP, a probabilistic algorithm that requires no more than t = e d ( 1 + δ ) ln ( n ) {\displaystyle t=ed(1+\delta )\ln(n)} tests to find up to d {\displaystyle d} defectives in n {\displaystyle n} items with a probability of error no more than n − δ {\displaystyle n^{-\delta }} . This is within a constant factor of the t = O ( d log 2 n ) {\displaystyle t=O(d\log _{2}n)} lower bound. Chan et al. (2011) also provided a generalisation of COMP to a simple noisy model, and similarly produced an explicit performance bound, which was again only a constant (dependent on the likelihood of a failed test) above the corresponding lower bound. In general, the number of tests required in the Bernoulli noise case is a constant factor larger than in the noiseless case. Aldridge, Baldassini and Johnson (2014) produced an extension of the COMP algorithm that added additional post-processing steps. They showed that the performance of this new algorithm, called DD, strictly exceeds that of COMP, and that DD is 'essentially optimal' in scenarios where d 2 ≥ n {\displaystyle d^{2}\geq n} , by comparing it to a hypothetical algorithm that defines a reasonable optimum. The performance of this hypothetical algorithm suggests that there is room for improvement when d 2 < n {\displaystyle d^{2}<n} , as well as suggesting how much improvement this might be.
Formalisation of combinatorial group testing This section formally defines the notions and terms relating to group testing.
The input vector, x = ( x 1 , x 2 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},x_{2},\dots ,x_{n})} , is defined to be a binary vector of length n {\displaystyle n} (that is, x ∈ { 0 , 1 } n {\displaystyle \mathbf {x} \in \{0,1\}^{n}} ), with the j-th item being called defective if and only if x j = 1 {\displaystyle x_{j}=1} . Further, any non-defective item is called a 'good' item.
x {\displaystyle \mathbf {x} } is intended to describe the (unknown) set of defective items. The key property of x {\displaystyle \mathbf {x} } is that it is an implicit input. That is to say, there is no direct knowledge of what the entries of x {\displaystyle \mathbf {x} } are, other than that which can be inferred via some series of 'tests'. This leads on to the next definition.
Let x {\displaystyle \mathbf {x} } be an input vector. A set, S ⊆ { 1 , 2 , … , n } {\displaystyle S\subseteq \{1,2,\dots ,n\}} is called a test. When testing is noiseless, the result of a test is positive when there exists j ∈ S {\displaystyle j\in S} such that x j = 1 {\displaystyle x_{j}=1} , and the result is negative otherwise. Therefore, the goal of group testing is to come up with a method for choosing a 'short' series of tests that allow x {\displaystyle \mathbf {x} } to be determined, either exactly or with a high degree of certainty.
A group-testing algorithm is said to make an error if it incorrectly labels an item (that is, labels any defective item as non-defective or vice versa). This is not the same thing as the result of a group test being incorrect. An algorithm is called zero-error if the probability that it makes an error is zero.
t ( d , n ) {\displaystyle t(d,n)} denotes the minimum number of tests required to always find d {\displaystyle d} defectives among n {\displaystyle n} items with zero probability of error by any group-testing algorithm. For the same quantity but with the restriction that the algorithm is non-adaptive, the notation t ¯ ( d , n ) {\displaystyle {\bar {t}}(d,n)} is used.
General bounds Since it is always possible to resort to individual testing by setting S j = { j } {\displaystyle S_{j}=\{j\}} for each 1 ≤ j ≤ n {\displaystyle 1\leq j\leq n} , it must be that that t ¯ ( d , n ) ≤ n {\displaystyle {\bar {t}}(d,n)\leq n} . Also, since any non-adaptive testing procedure can be written as an adaptive algorithm by simply performing all the tests without regard to their outcome, t ( d , n ) ≤ t ¯ ( d , n ) {\displaystyle t(d,n)\leq {\bar {t}}(d,n)} . Finally, when 0 ≠ d ≠ n {\displaystyle 0\neq d\neq n} , there is at least one item whose defectiveness must be determined (by at least one test), and so 1 ≤ t ( d , n ) {\displaystyle 1\leq t(d,n)} . In summary (when assuming 0 ≠ d ≠ n {\displaystyle 0\neq d\neq n} ), 1 ≤ t ( d , n ) ≤ t ¯ ( d , n ) ≤ n {\displaystyle 1\leq t(d,n)\leq {\bar {t}}(d,n)\leq n} .
Information lower bound A lower bound on the number of tests needed can be described using the notion of sample space, denoted S {\displaystyle {\mathcal {S}}} , which is simply the set of possible placements of defectives. For any group testing problem with sample space S {\displaystyle {\mathcal {S}}} and any group-testing algorithm, it can be shown that t ≥ ⌈ log 2 | S | ⌉ {\displaystyle t\geq \lceil \log _{2}{|{\mathcal {S}}|}\rceil } , where t {\displaystyle t} is the minimum number of tests required to identify all defectives with a zero probability of error. This is called the information lower bound. This bound is derived from the fact that after each test, S {\displaystyle {\mathcal {S}}} is split into two disjoint subsets, each corresponding to one of the two possible outcomes of the test. However, the information lower bound itself is usually unachievable, even for small problems. This is because the splitting of S {\displaystyle {\mathcal {S}}} is not arbitrary, since it must be realisable by some test. 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. For any group-testing algorithm that performs t {\displaystyle t} tests, the probability of success, P ( success ) {\displaystyle \mathbb {P} ({\textrm {success}})} , satisfies P ( success ) ≤ t / log 2 ( n d ) {\displaystyle \mathbb {P} ({\textrm {success}})\leq t/\log _{2}{n \choose d}} . This can be strengthened to: P ( success ) ≤ 2 t ( n d ) {\displaystyle \mathbb {P} ({\textrm {success}})\leq {\frac {2^{t}}{n \choose d}}} .
Representation of non-adaptive algorithms
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.
Suppose a non-adaptive group testing procedure for n {\displaystyle n} items consists of the tests S 1 , S 2 , … , S t {\displaystyle S_{1},S_{2},\dots ,S_{t}} for some t ∈ N ≥ 0 {\displaystyle t\in \mathbb {N} _{\geq 0}} . The testing matrix for this scheme is the t × n {\displaystyle t\times n} binary matrix, M {\displaystyle M} , where ( M ) i j = 1 {\displaystyle (M)_{ij}=1} if and only if j ∈ S i {\displaystyle j\in S_{i}} (and is zero otherwise). Thus each column of M {\displaystyle M} represents an item and each row represents a test, with a 1 {\displaystyle 1} in the ( i , j ) -th {\displaystyle (i,j){\textrm {-th}}} entry indicating that the i -th {\displaystyle i{\textrm {-th}}} test included the j -th {\displaystyle j{\textrm {-th}}} item and a 0 {\displaystyle 0} indicating otherwise. As well as the vector x {\displaystyle \mathbf {x} } (of length n {\displaystyle n} ) that describes the unknown defective set, it is common to introduce the result vector, which describes the results of each test.
Let t {\displaystyle t} be the number of tests performed by a non-adaptive algorithm. The result vector, y = ( y 1 , y 2 , … , y t ) {\displaystyle \mathbf {y} =(y_{1},y_{2},\dots ,y_{t})} , is a binary vector of length t {\displaystyle t} (that is, y ∈ { 0 , 1 } t {\displaystyle \mathbf {y} \in \{0,1\}^{t}} ) such that y i = 1 {\displaystyle y_{i}=1} if and only if the result of the i -th {\displaystyle i{\textrm {-th}}} test was positive (i.e. contained at least one defective). With these definitions, the non-adaptive problem can be reframed as follows: first a testing matrix is chosen, M {\displaystyle M} , after which the vector y {\displaystyle \mathbf {y} } is returned. Then the problem is to analyse y {\displaystyle \mathbf {y} } to find some estimate for x {\displaystyle \mathbf {x} } . In the simplest noisy case, where there is a constant probability, q {\displaystyle q} , that a group test will have an erroneous result, one considers a random binary vector, v {\displaystyle \mathbf {v} } , where each entry has a probability q {\displaystyle q} of being 1 {\displaystyle 1} , and is 0 {\displaystyle 0} otherwise. The vector that is returned is then y ^ = y + v {\displaystyle {\hat {\mathbf {y} }}=\mathbf {y} +\mathbf {v} } , with the usual addition on ( Z / 2 Z ) n {\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}} (equivalently this is the element-wise XOR operation). A noisy algorithm must estimate x {\displaystyle \mathbf {x} } using y ^ {\displaystyle {\hat {\mathbf {y} }}} (that is, without direct knowledge of y {\displaystyle \mathbf {y} } ).
Bounds for non-adaptive algorithms The matrix representation makes it possible to prove some bounds on non-adaptive group testing. The approach mirrors that of many deterministic designs, where d {\displaystyle d} -separable matrices are considered, as defined below.
A binary matrix, M {\displaystyle M} , is called d {\displaystyle d} -separable if every Boolean sum (logical OR) of any d {\displaystyle d} of its columns is distinct. Additionally, the notation d ¯ {\displaystyle {\bar {d}}} -separable indicates that every sum of any of up to d {\displaystyle d} of M {\displaystyle M} 's columns is distinct. (This is not the same as M {\displaystyle M} being k {\displaystyle k} -separable for every k ≤ d {\displaystyle k\leq d} .) When M {\displaystyle M} is a testing matrix, the property of being d {\displaystyle d} -separable ( d ¯ {\displaystyle {\bar {d}}} -separable) is equivalent to being able to distinguish between (up to) d {\displaystyle d} defectives. However, it does not guarantee that this will be straightforward. A stronger property, called disjunctness does.
A binary matrix, M {\displaystyle M} is called d {\displaystyle d} -disjunct if the Boolean sum of any d {\displaystyle d} columns does not contain any other column. (In this context, a column A is said to contain a column B if for every index where B has a 1, A also has a 1.) A useful property of d {\displaystyle d} -disjunct testing matrices is that, with up to d {\displaystyle d} defectives, every non-defective item will appear in at least one test whose outcome is negative. This means there is a simple procedure for finding the defectives: just remove every item that appears in a negative test. Using the properties of d {\displaystyle d} -separable and d {\displaystyle d} -disjunct matrices the following can be shown for the problem of identifying d {\displaystyle d} defectives among n {\displaystyle n} total items.
The number of tests needed for an asymptotically small average probability of error scales as O ( d log 2 n ) {\displaystyle O(d\log _{2}n)} . The number of tests needed for an asymptotically small maximum probability of error scales as O ( d 2 log 2 n ) {\displaystyle O(d^{2}\log _{2}n)} . The number of tests needed for a zero probability of error scales as O ( d 2 log 2 n log 2 d ) {\displaystyle O\left({\frac {d^{2}\log _{2}n}{\log _{2}d}}\right)} .
Generalised binary-splitting algorithm
The generalised binary-splitting algorithm is an essentially-optimal adaptive group-testing algorithm that finds d {\displaystyle d} or fewer defectives among