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| In [[combinatorics|combinatorial mathematics]], '''group testing''' refers to any procedure which breaks up the task of locating elements of a set which have certain properties into tests on subsets ("groups") rather than on individual elements. A familiar example of this type of technique is the false coin problem of [[recreational mathematics]]. In this problem there are ''n'' coins and one of them is false, weighing less than a real coin. The objective is to find the false coin, using a [[balance scale]], in the fewest number of weighings. By repeatedly dividing the coins in half and comparing the two halves, the false coin can be found quickly as it is always in the lighter half.<ref>A bit more precisely – if there are an odd number of coins to be weighed, pick one to put aside and divide the rest into two equal piles. If the two piles have equal weight, the bad coin is the one put aside, otherwise the one put aside was good and no longer has to be tested.</ref>
| | If you present photography effectively, it helps you look much more properly at the globe around you. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. * A community forum for debate of the product together with some other customers in the comments spot. Keep reading for some great Word - Press ideas you can start using today. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. Setting Up Your Business Online Using Free Wordpress Websites. You or your web designer can customize it as per your specific needs. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. <br><br>Word - Press installation is very easy and hassle free. Find more information about Design To Wordpress here. To do this, you should link your posts to other relevant posts that you've created. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. If you loved this post and you want to receive more details regarding [http://gamesforrelax.com/wordpress_dropbox_backup_778350 wordpress dropbox backup] kindly visit our own page. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. In fact portfolio Word - Press themes is a smooth and attractive but considerably flawed Word - Press theme in creating simpler to the photographers or designers to develop a specific internet site showcasing their most current perform since it appear modern-day and has fantastic typography and large photographs which would develop an attractive wanting portfolio internet site. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. And, it is better that you leave it on for the duration you are writing plugin code. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account. |
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| Schemes for carrying out such 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'''.
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| ==Background==
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| Robert Dorfman's paper in 1943 introduced the field of (Combinatorial) Group Testing. 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. The objective was to weed out all syphilitic men called up for induction. However, syphilis testing back then was expensive and testing every soldier individually would have been very cost heavy and inefficient.
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| A basic breakdown of a '''test''' is:
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| * Draw sample from a given individual
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| * Perform required tests
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| * Determine presence or absence of syphilis
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| Say we have <math>n</math> soldiers, then this method of testing leads to <math>n</math> tests. If we have 70-75% of the people infected then the method of individual testing would be reasonable. Our goal however, is to achieve effective testing in the more likely scenario where it does not make sense to test 100,000 people to get (say) 10 positives.
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| The feasibility of a more effective testing scheme hinges on the following property. We can combine blood samples and test a combined sample together to check if at least one soldier has syphilis.
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| Modern interest in these testing schemes has been rekindled by the Human Genome Project.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg. 574, Section 46: Pooling Designs}}</ref>
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| ==Formalization of the problem==
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| We now formalize the group testing problem abstractly.
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| <math>Input:</math> The total number of soldiers <math>n</math>, an [[upper bound]] on the number of infected soldiers <math>d</math>. The (unknown) information about which soldier is infected described as a [[Tuple|vector]] <math>\mathbf{x}= (x_1, x_2, ..., x_n)</math> where <math>x_i = 1</math> if the item <math>i</math> is infected else <math>x_i = 0</math>.
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| The ''Hamming Weight'' of <math>\mathbf{x}</math> is defined as the number of <math>1's</math> in <math>x</math>. Hence, <math>|x| \leq d</math> where <math>|x|</math> is the [[Hamming weight]]. The vector <math>\mathbf{x}</math> is an implicit input since we do not know the positions of <math>1's</math> in the input. The only way to find out is to run the tests.
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| ===Formal notion of a Test===
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| A '''query/test''' <math>S</math> is a subset of <math>[n]</math>. The answer to the query <math> S \subseteq [n] </math> is defined as follows:
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| <math>A(S) = \begin{cases} 1, \mbox{ if } \displaystyle\sum_{k\in S} x_k \geq 1\\ 0, \mbox{ otherwise.} \end{cases}</math> | |
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| Note that the addition operation used by the summation is the logical-<math>OR</math>, i.e.
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| <math>A(S) = \displaystyle\bigvee_{i\in S} x_i</math>.
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| ===Goal===
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| Compute <math>\mathbf{x}</math> and minimize the number of tests required to determine <math>\mathbf{x}</math>
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| The question boils down to one of '''''Combinatorial Searching'''''. Combinatorial searching in general can be explained as follows: Say you have a set of <math>n</math> variables and each of these can take on <math>m</math> possible values. So, finding possible solutions that match a certain constraint is a problem of combinatorial searching. The major problem with such questions is that the solution can grow [[Exponential growth|exponentially]] in the size of the input. Here, we have no direct questions or answers. Any piece of information can only be obtained using an indirect query.
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| ===Definition===
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| <math>t(d,n):</math> Given a set of <math>n</math> items with <math>d</math> defects, the minimum number of tests that one would have to make to detect all the defective items is defined as <math>t(d,n)</math>.
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| Consider the case when only one person in the group will test positive. Then if we tested in the naive way, in the best case we would at least have to test the first person to find out if he/she is infected. However, in the worst case one might have to end up testing the entire group and only the last person we test will turn out to really be the one who was infected. Hence, <math>1 \leq t(d, n) \leq n</math>
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| ===Testing Methods===
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| There are two basic principles via which the testing may be carried out:
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| # ''Adaptive Group Testing'' is where we test a given subset of items and, we get the answer from the test. We then base the next test on the outcome of the current test.
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| # ''Non-adaptive Group Testing'' on the other hand is when all the tests to be performed are decided ''a priori''.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg. 631, Section 56.4}}</ref>
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| ====Definition====
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| <math>t^a(d,n):</math> Given a set of <math>n</math> items with <math>d</math> defects, <math>t^a(d,n):</math> is defined as the number of adaptive tests that one would have to make to detect all the defective items.
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| One should note that in the case of group testing for the Syphilis problem, non-adaptive group testing is crucial. This is because the soldiers might be spread out geographically and adaptive group testing will need a lot of co-ordination.
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| ===Mathematical representation of the set of non-adaptive tests===
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| For, <math>S \subseteq [n]</math>, define <math> \chi_i \in \{ 0,1 \}^n</math> such that <math> i \in S \Leftrightarrow \chi_s(i) = 1</math>. <math>M</math> is a <math>t \times n</math> matrix of <math>\chi_i</math>. <math>\mathbf{x}</math> is the input vector transposed and <math>\mathbf{r}</math> is the resultant. The construction is based on the grounds that for non-adaptive testing with <math>t</math> tests is represented by a <math>t-</math>subset <math>S_i \subseteq [n] (1 \leq i \leq t)</math>. <math>\chi_i</math> for a given <math>i</math> is the <math>i^{th}</math> test. <math>M</math> test matrix where <math>m_{i,j}</math> is one if for the <math>i^{th}</math> test, <math> j \in S</math>. Note that here multiplication is logical AND (<math>\bigwedge</math>) and addition is logical OR (<math>\bigvee</math>). Then, <math>M \times \mathbf{x} = \mathbf{r}</math> where <math>\mathbf{r}</math> is the resultant of the matrix multiplication. To think of this in terms of testing, it is helpful to visualize matrix multiplication. Here, <math>\mathbf{r}</math> will have a 1 in position <math>i</math> if and only if there was a <math>1</math> in that position in both <math>M</math> and <math>\mathbf{x}</math> i.e. if that person was tested with that particular group and if he tested out to be positive.
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| <math>M =
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| \begin{pmatrix} m_{1,1} \cdots m_{1,n} \\ \vdots \ddots \vdots \\ m_{t,1} \cdots m_{t,n} \end{pmatrix}</math>
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| <math>\mathbf{x} = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \mathbf{r} = \begin{pmatrix} r_1 \\ \vdots \\ r_t \end{pmatrix}</math>
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| ===Bounds for testing on <math>t^a(d,n)</math> and <math>t(d,n)</math>===
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| <math> 1 \leq t^a(d,n) \leq t(d,n) \leq n </math>
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| The reason for <math>t^a(d,n) \leq t(d,n)</math> is due to the fact that any non-adaptive test can be performed by an adaptive test by running all of the tests in the first step of the adaptive test. Adaptive tests can be more efficient than non-adaptive tests since the test can be changed after certain things are discovered.
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| ====Lower bound on <math>t^a(d,n) </math>====
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| Fix a valid group testing scheme with <math>t</math> tests. Now, for two distinct vectors <math>\mathbf{x} </math> and <math>\mathbf{y} </math> where <math>|\mathbf{x}|, |\mathbf{y}| \leq d</math>, the resulting vectors will not be the same i.e. <math>\mathbf{r(x)} \neq \mathbf{r(y)}</math>. Here <math>\mathbf{r(x)}</math> is the resultant vector when <math>\mathbf{x}</math>. This is because, two valid inputs will never give us the same result. If this ever happened, then we would always have an error in finding both <math>\mathbf{x} </math> and <math>\mathbf{y}</math>. This gives us that the total number of distinct results is the volume of a Hamming Ball of radius <math>d</math>, centered about <math>n</math> i.e. <math>Vol_2(d,n)</math>. However, for <math>t</math> bits, the total number of possible distinct vectors is <math>2^t</math>. Hence, <math>2^t \geq Vol_2(d,n)</math>. Taking the <math>\log</math> on both sides gives us <math>t \geq \log\{Vol_2(d,n)\}</math>.
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| Now, <math>Vol_2(d,n) \geq {n \choose d} \geq (\frac{n}{d})^d</math>. Therefore, we will end up having to perform a minimum of <math>d\log{\frac{n}{d}}</math> tests.
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| Thus we have proved, <math>t^a(d,n) \geq d\log\frac{n}{d}</math>
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| ====Upper bound on <math>t^a(d,n) </math>====
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| <math> t^a(d,n) \leq O(d\log{n}) </math>.
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| Since we know that the upper bound on the number of positives is <math>d</math>, we run a binary search at most <math>d</math> times or until there are no more values to be found. To simplify the problem we try to give a testing sccheme that uses <math>O(\log{n})</math> adaptive tests to figure out a <math>i</math> such that <math>x_i = 1</math>. The related problem is solved by splitting <math>[n]</math> in two halves and querying to find a <math>1</math> in one of those and then proceeding recursively to find the exact position in the half where the query returned a <math>1</math>. This will take <math>2\lceil\log{n}\rceil </math> time or if the first query is performed on the whole set, it will take <math>\lceil\log{n}\rceil +1</math>. Once a <math>1</math> is found, the search is then repeated after removing the <math>i^{th}</math> co-ordinate. This can be done at most <math>d</math> times. This justifies the running time of <math>O(d\log{n}) </math>.
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| For a full proof and an algorithm for the problem refer to: [http://www.cse.buffalo.edu/~atri/courses/coding-theory/spr10/lectures/lect11.pdf CSE545 at the University at Buffalo]
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| ====Upper bound on <math>t(1,n)</math>====
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| <math>t(1,n) \leq \lceil\log{n}\rceil</math>
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| This upper bound is for the special case where <math>d = 1</math> i.e. there is a maximum of 1 positive. In this case, the matrix multiplication gets simplified and the resultant <math>\mathbf{r}</math> represents the binary representation of <math>i</math> for test <math>i</math>. This gives a lower bound of <math>\lceil\log{n}\rceil</math>. Note that decoding becomes trivial because the binary representation of <math>i</math> gives us the location directly. The group test matrix here is just the parity check matrix <math>H_m</math> for the <math>[2^m - 1, 2^m-m-1, 3]</math> [[Hamming code]].
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| Thus as the upper and lower bounds are the same, we have a tight bound for <math>t(d,n)</math> when <math> d = 1</math>. Such tight bounds are not known for general <math>d</math>.
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| ==== Upper Bounds for Non-Adaptive Group Testing ====
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| For non-adaptive group testing upper bounds we shift focus toward [[disjunct matrix|disjunct matrices]]. Disjunct matrices have been used for many of the bounds because of their nice properties. Through use of different constructions of disjunct matrices it has been shown that <math>\Omega(\frac{d^2}{\log{d}}\log{n}) \leq t(d,n)</math>. Also for upper bounds we currently have that (i) <math>t(d,n) \leq \mathcal{O}(d^2 \log{n}) </math>(explicit construction) and (ii) <math>t(d,n) \leq \mathcal{O}(d^2 \log^2{n}) </math>(strongly explicit construction). It is good to note that the current known lower bound for <math>t(d,n)</math> is already a <math>\frac{d}{\log{d}}</math> factor larger than the upper bound for <math>t^a(d,n)</math>. Another thing to note is that give the smallest upper bound and biggest lower bound they are only off by a factor of <math>\frac{1}{\log{d}}</math> which is fairly small.
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| == See also ==
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| * [[Disjunct Matrix]]
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| * [[Robert Dorfman]]
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| * [[Concatenated error correction codes]]
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| * [[Hamming weight]]
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| * [[Hamming code]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * Atri Rudra's course on Error Correcting Codes: Combinatorics, Algorithms, and Applications (Spring 2007), Lectures [http://www.cse.buffalo.edu/~atri/courses/coding-theory/lectures/lect7.pdf 7].
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| * Atri Rudra's course on Error Correcting Codes: Combinatorics, Algorithms, and Applications (Spring 2010), Lectures [http://www.cse.buffalo.edu/~atri/courses/coding-theory/spr10/lectures/lect10.pdf 10], [http://www.cse.buffalo.edu/~atri/courses/coding-theory/spr10/lectures/lect11.pdf 11], [http://www.cse.buffalo.edu/~atri/courses/coding-theory/spr10/lectures/lect28.pdf 28], [http://www.cse.buffalo.edu/~atri/courses/coding-theory/spr10/lectures/lect29.pdf 29]
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| * {{citation|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|year=2007|publisher=Chapman & Hall/ CRC|location=Boca Raton|isbn=1-58488-506-8|edition=2nd Edition}}
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| * Dorfman, R. The Detection of Defective Members of Large Populations. The Annals of Mathematical Statistics, 14(4), 436-440. Retrieved from [http://www.jstor.org/pss/2235930]
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| * Du, D., & Hwang, F. (2006). Pooling Designs and Nonadaptive Group Testing. Boston: Twayne Publishers.
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| * Ely Porat, Amir Rothschild: Explicit Non-adaptive Combinatorial Group Testing Schemes. ICALP (1) 2008: 748-759
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| [[Category:Combinatorics]]
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| [[Category:Design of experiments]]
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