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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>
| | CMS provides the best platform to create websites that fulfill all the specifications of SEO. It is very easy to customize plugins according to the needs of a particular business. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. After age 35, 18% of pregnancies will end in miscarriage. Our skilled expertise, skillfulness and excellence have been well known all across the world. Have you heard about niche marketing and advertising. <br><br>You can add keywords but it is best to leave this alone. Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. The Pakistani culture is in demand of a main surgical treatment. <br><br>Yet, overall, less than 1% of websites presently have mobile versions of their websites. I'm a large fan of using Word - Press to create pretty much any sort of web page. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. If you adored this article and you would such as to get more details relating to [http://mmservice.dk/wordpress_backup_51559 wordpress backup] kindly see the web-page. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
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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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