Lecturer: Huilan Chang Department of Applied Mathematics, National University of Kaohsiung Combinatorial Group Testing 組合群試設計 Fall, 2013

Lecturer: Huilan ChangDepartment of Applied Mathematics,

National University of Kaohsiung

Combinatorial Group Testing 組合群試設計

Fall, 2013

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Course Homepage: http://140.127.223.1/hchang/102I/PoolingDesign.htm

Grading Policy: Participation(20%)In-class Discussion(20%)In-class presentation and Final project (optional) (60%)( 一人報一篇 or 兩人報兩篇 )

學習目標了解此領域的研究 , 進而解決問題 !

Introduction of the course

Classical Group TestingA set of n items is given.

Each item is either positive or negative but we don’t know who are positive!

Known Information: there are at most d positive items.Called (n, d) model.

Goal: identify all positive ones by the minimum number of group tests.

Group test (or pool) : a subset S of items. We can apply a test on it. The outcome is positive iff S contains at least one positive item.

A Glimpse of Group Testing

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Classical Group TestingExample. Suppose there is at most one positive item.


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3

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1

5

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negative pool

positive poolpositive pool

Which one is positive?

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History: World War II 1939-45. Dorfman suggested to group the blood samples.(ref. R. Dorfman, The detection of defective members of large populations, Ann. Math. Statist. 14 (1943) 436-440)


Applicationsblood testing (since 1943), chemical leakage testing, electric

shorting detection, DNA library screening, compressed sensing, fingerprinting, network security…

Different applications correspond to various models of group testing where different codes may be applied to solve it (nonadaptive approach).


An application in molecular biologyDNA sequencing: the process of determining the precise order of nucleotides

within a DNA molecule.

General technique: Copy the genome and cut them into pieces (called clones) by different

restriction enzyme.Reconstruct the clones.Reconstruct the target sequence.


Physical mappingSTS (sequence-tagged sites) is a short DNA that appears uniquely in the

target sequence and clones.We don’t know the order of STSs on the target segment!!

Idea: If two clones share an STS, then they mush be adjacent.Rearrange: Delete cones whose STS-set is a subset of another clone and

redundant ones.Reconstruct the target sequence by the overlapping graph.

Overlapping graphAB—BC—CD—DE—EFG

Result: Hamiltonian path ABCDEFG

Target sequence A B C D E F G (STSs)

First cut A B C D E F G

Second cut A B C D E F G

Third cut A B C D E F G


How are the clones reconstructed? Suppose we have seven STSs: A, B, C, D, E, F, G. Identify the STS-set for each clone by one STS at a time.

E

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In this example, we cut the target sequence three times so for any STS, there are at most three segments having it.It corresponds to (10, 3) clone model.A test is accomplished by a probe ( 探針 ) for a STS.

Target sequence A B C D E F G

First cut A B C D E F G

Second cut A B C D E F G

Third cut A B C D E F G


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Various models:Classical group testing: the number of positives is at most d.Competitive group testing: no information on the number of

positives.

Inhibitor model (Farach et al. 1997)Complex model (Torney 1999)Inhibitor-Complex model (Chang et al. 2010)

Threshold group testing (Damaschke, 2006)Majority group testing (Ahlswede et al. 2010)Density-based group testing (Gerbner et al. arXiv 2012)

Graph-constraint group testing (Cheraghchi et al. 2012), e.g., Consecutive group testing (Colbourn,1999)

.....


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Textbook:My T. Thai, Group Testing Theory in Network Security

-Advanced Solution, 2012. ( 有電子書 )D. Z. Du and F. K. Hwang, Pooling Designs and Nonadaptive

Group Testing - Important Tools for DNA Sequencing, World Scientific, 2006.

D. Z. Du and F. K. Hwang, Combinatorial Group Testing and Its Applications (2nd Edit.), World Scientific, 2000.

M. Aigner. Combinatorial Search. John Wiley and Sons, 1988.

更重要的 reference: Journal Articles!


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What to do:1. Provide algorithms to find all positive items.

Types of algorithms:Sequential algorithm conducts the tests one by one.

e.g. binary search algorithm. Nonadaptive algorithm (pooling design):

o prepare all pools according to the known information then apply tests on them simultaneouslyo recover positives from the testing outcomes.

(decoding algorithm)


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Sequential algorithm needs less number of tests, but longer time. Nonadaptive design needs more tests, but shorter time.k-stage algorithm: stages are sequential and all tests in a

stage are nonadaptive

2. Study the lower bound!


SequentialFor the (n,d) model,

a binary search algorithm can identify a positive item

in at most tests. tests suffice for the (n,d) model.

The information-theoretical lower bound:

When d=1, binary search algorithm is optimal. When d>1, no algorithm matching the information lower bound is found!!

1 2 3 4


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negative positive

1 2 4

negative

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Nonadaptive algorithm 例子：假設「有 40 個東西編號為 1, 2, …, 40 ，其中一個是正的」。根據此訊息準備以下六個 pools (or tests) ，不管哪個東西是正的，我都可以從測驗結果找出正的那一個。

因此 ,nonadaptive algorithm 和 coding theory 、 design theory… 有重大關聯 !


NonadaptiveThe only information: there are 7 items and one of them

is positive. ((7,1) model)

1 2 3 4 5 6 7P1 1 0 1 0 1 0 1P2 0 1 1 0 0 1 1 P3 0 0 0 1 1 1 1

Outcome011

♦View columns as sets of poolsEx. First column={1} 代表第一個東西在第一個測驗被測到；Third column={1,2} 代表第三個東西在第一和第二個測驗被測到。試解釋下列 column 的運算 :• union of columns,• intersection of columns, • one column covers another column.


♦ P1={1,3,5,7} is a negative pool.

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NonadaptiveThe only information: there are 7 items and one of them

is positive. ((7,1) model)

Notice that all columns are distinct!How about decoding? It depends on the property of the

matrix!

1 2 3 4 5 6 7P1 1 0 1 0 1 0 1P2 0 1 1 0 0 1 1 P3 0 0 0 1 1 1 1

Outcome011


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Nonadaptive approach:

Understand Model Pooling Design (Matrix with certain property) + Decoding Algorithm Construction of such matrices!

Example: (n, 1) model___________________ +__________________

________________________________________


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Suppose: n items and exactly d positives.A nonadaptive design:

How to decode:

Find the set of d columns whose union is the testing outcome!


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Classical group testing: n items and at most d positives.A nonadaptive design:

How to decode:

Find the set of at most d columns whose union is the testing outcome!


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Classical group testing: n items and at most d positives.A nonadaptive design: A matrix is d-disjunct if for any d+1 columns C0, C1, …, Cd+1,

there is one row intersecting C0, but none of C1, …, Cd+1.

How to decode:

Every negative item must appear in a pool without positives and thus it appears in a negative pool!


d

… 1 0 0 0 0 0 0 ...

C0

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Example. A 9×12 2-disjunct matrix constructed by block design.


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Any nonadaptive algorithm must use a number of

tests (Erdös, Frankl and Füredi, 1985).

The number of rows of a d-disjunct with n columns has upper bound:

Deterministic construction (Hwang and Sós 1987).Probabilistic construction (Cheng and Du 2008).


)log( 2 ndO

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A. 從 Classical group testing 開始介紹：1. Sequential algorithm and lower bound (skipped)

2. Nonadaptive algorithm (several papers):Design and decoding (d-separable, d-disjunct…)Construction: deterministic construction and probabilistic construction

3. Optimal trivial 2-stage algorithm:Design and decodingConstruction

接下來… .

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B. Threshold group testing v.s. Group testing with consecutive positivesProject Chair: 蔡宜霖

1. Threshold group testing (Damaschke 2006) 1. P. Damaschke, Threshold group testing, In: General Theory of

Information Transfer and Combinatorics, Lect. Notes Comput. Sci. 4123 (2006) 707-718. *Model description/Sequential algorithm/Lower bound

2. H.-B. Chen and H.-L. Fu, Nonadaptive algorithms for threshold group testing, Discrete Appl. Math. 157 (2009) 1581-1585.*Nonadaptive approach (in disjunct sense)

3. M. Cheraghchi, Improved constructions for non-adaptive threshold group testing, Algorithmica (2013) (DOI 10.1007/s00453-013-9754-7).***Nonadaptive approach (in separable sense)

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4. A. D’yachkov, V. Rykov, C. Deppe, V. Lebedev, Superimposed Codes and Threshold Group Testing, Info. Theory, Combin., and Search Theory, LNCS, 7777 (2013) 509-533.** 可以期待 !

5. -Near-Optimal Stochastic Threshold Group Testing-Stochastic Threshold Group Testing*** 什麼？有這種東西？ !

自選文章 ! ( 老師要先審核過 !)

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B. Threshold group testing v.s. Group testing with consecutive positives

2. Group testing with consecutive positives (Damaschke 2006) 1. C. J. Colbourn, Group testing for consecutive positives, Ann.

Combin. 3 (1999) 37-41.*Model description/Nonadaptive algorithm/Lower bound

2. J. S.-T. Juan and G. J. Chang, Adaptive group testing for consecutive positives, Discrete Math., 308 (2008) 1124-1129.*Sequential algorithm

3. M. Müller and M. Jimbo, Consecutive positive detectable matrices and group testing for consecutive positives, Discrete Math, 279 (2004) 369–381.*Nonadaptive, Optimal design for d=2.

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4. *M. Müller and M. Jimbo, Cyclic sequences of k-subsets with distinct consecutive unions, Discrete Math. 308 (2008) 457–464.

B. Threshold group testing v.s. Group testing with consecutive positives

1. Threshold Group Testing with Consecutive Positives (Chang, Chiu, Tsai, 2013)-Sequential-Nonadaptive

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C. Pooling Design with Error-correctingProject Chair: 蔡一慈

1. *A.J. Macula, A simple construction of d-disjunct matrices with certain constant weights, Discrete Math. 162 (1996) 311–312.

2. *J. Guo, K. Wang, A construction of pooling designs with high degree of error correction, J. Combin. Theory Ser. A 118 (2011) 2056–2058.

3. A.G. D’yachkov, F.K. Hwang, A.J. Macula, P.A. Vilenkin, C. Weng, A construction of pooling designs with some happy surprises, J. Comput. Biol. 12 (2005) 1127–1134.

*H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance pooling designs, Discrete Math. 243 (2002) 161–170.

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4. J. Nan, J. Guo, New error-correcting pooling designs associated with finite vector spaces, J. Comb. Optim. 20 (2010) 96–100.

*J. Guo, K. Wang, Pooling designs with surprisingly high degree of error correction in a finite vector space, Discrete App. Math. 160 (2012) 2172–2176.

5. Generalizations of Guo and Wang’s disjunct matrices!

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6. **T. Huang, K.Wang, C.-W. Weng, Pooling spaces associated with finite geometry, European J of Combin, 29 (2008)1483–1491.

7. * P. Zhao, K. Diao, K. Wang, A generalization of Macula’s disjunct matrices, Journal of Comb Optim, 22 (2011) 495-498.

8. **S. Gaoa. Z. Lib, J. Yuc, X. Gaoc, W. Wu, DNA library screening, pooling design and unitary spaces, Theoretical Comp Sci, 412 (2011) 217–224.

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9. ***Z. Li, T. Huang, S. Gao,Two error-correcting pooling designs from symplectic spaces over a finite field, Linear Algebra and its Appl. 433 (2010) 1138–1147.

10. *Y. Bai, T. Huang and K. Wang, Error-correcting pooling designs associated with some distance regular graphs, Discrete Applied Mathematics 157 (2009), 3038-3045.

自選文章 ! ( 老師要先審核過 !)

教學計畫

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Lecturer: Huilan Chang Department of Applied Mathematics, National University of Kaohsiung Combinatorial Group Testing 組合群試設計 Fall, 2013