Threshold Group Testing with Consecutive Positives

Advisor : Huilan Chang

Student : Yi-Lin Tsai

Department of Applied MathematicsNational University of Kaohsiung

2014/08/02

Outline

2

Introduction• Group testing

• Group testing with consecutive positives

• Threshold group testingMain result• Sequential algorithm for T.G.T.C

• Nonadaptive algorithm for T.G.T.C

Concluding

Reference

Classical group testing

• Given a set of items, each is either positive

or negative, and a set of at most positives.

• Goal : identify all positives by group test.

• Group Test : a test on a subset .

3

Positive outcome: contains at least one positive item.

positive negative

Types of algorithm

• Sequential algorithm : A test can be specified after the previous test outcome.

• Nonadaptive algorithm : All test are specified beforehand and are conducted simultaneously.

4

[1 1 0 1 0 1 0 0 00 1 0 0 1 1 0 1 01 0 1 0 0 1 1 1 01 0 0 1 0 0 1 0 10 1 1 0 1 0 0 1 01 1 0 0 1 0 1 1 0

]𝑐1𝑐2𝑐3𝑐4𝑐5𝑐6𝑐7𝑐8𝑐9

𝑡1𝑡 2

𝑡 3

𝑡 4

𝑡5𝑡 6

items

tests

Types of algorithm

• Sequential algorithm : A test can be specified after the previous test outcome.

• Nonadaptive algorithm : All test are specified beforehand and are conducted simultaneously.

4

[1 1 0 1 0 1 0 0 00 1 0 0 1 1 0 1 01 0 1 0 0 1 1 1 01 0 0 1 0 0 1 0 10 1 1 0 1 0 0 1 01 1 0 0 1 0 1 1 0

]𝑐1𝑐2𝑐3𝑐4𝑐5𝑐6𝑐7𝑐8𝑐9

𝑡1𝑡 2

𝑡 3

𝑡 4

𝑡5𝑡 6

items

tests

1

001

0

0

Outcome vector

Consecutive model

• is a set of items with the linear order for .

• : is a set of positive items which is consecutive (under the ordering ), and contains at most items.

• Test : choose arbitrary subset of .

5

• Balding and Torney (1997) and Colbourn (1999) first studied this model.• Colbourn (1999)

• Mller and Jimbo (2004)

• Juan and Chang (2008)

Consecutive model

6

sequential : nonadaptive :

nonadaptive :

sequential :

𝐥𝐨𝐠𝟐𝒏+𝐥𝐨𝐠𝟐𝒅+𝒄 𝐥𝐨𝐠𝟐 ⌈𝒏

𝒅−𝟏⌉+𝟐𝒅+𝟏

𝐥𝐨𝐠𝟐 ⌈𝒏

𝒅−𝟏⌉+𝟐𝒅−𝟏

, for

Lower bound :

Threshold group testing

7

• Peter Damaschke (2006)

𝒖𝒍

upper threshold

negative

lower threshold

positivearbitrary answer

Threshold group testing

7

𝒖𝒍

• Peter Damaschke (2006)

• If then we can find all positives.If then we can only find a -approximate set.

upper thresholdlower threshold

𝒈

• A set is called -approximate

if and .

EX1 Let .

EX2 The classical group testing is the case of .

Threshold group testing

a b

c d e

-approximate set

8

9

Group testing with consecutive

positives

Threshold group testing

Threshold group testing with consecutive positives

Our work

Our work

Threshold group testing with consecutive positives

• Lower bound

.

• Sequential algorithm

.

• Nonadaptive algorithm

and

decoding complexity : .

10

Main result

Sequential algorithm

Nonadaptive algorithm

Sequential algo. for T.G.T.C

12

Recall :

It is usually assumed that .

…

n items

at most positives

Sequential algo. for T.G.T.CInformation-theoretic lower bound :

Proposition (Chang and Tsai, 2014)

If , then the number of group tests

required to identify all positive items from is

at least

.

13

Sequential algo. for T.G.T.COur job :

Provide an algorithm to locate all positive items from linear

order and compare with the lower bound.

…

14

at most positives

Sequential algo. for T.G.T.C

…

14

at most positives

Our job :

Provide an algorithm to locate all positive items from linear

order and compare with the lower bound.

Sequential algo. for T.G.T.C

…

min max

We start with the case gap .

14

at most positives

Our job :

Provide an algorithm to locate all positive items from linear

order and compare with the lower bound.

.

.

Threshold without gap

15

Theorem 1 (Chang and Tsai, 2014)

For gap-free T.G.T.C, all positives can be identified in

tests.

Threshold without gapProof of Theorem 1

16

𝓝 • First partition into parts of consecutive items andadd some dummy negative items to the last part. dummy

items

𝑋 1 𝑋 3𝑋 2 𝑋 ⌈ 𝑛/𝑢⌉𝑋 4 ⋯⋯⋯⋯• Let .

• Goal : find min.𝓝𝑋 1 𝑋 3𝑋 2 𝑋 ⌈ 𝑛/𝑢⌉𝑋 4 ⋯⋯⋯⋯𝑋 5

Algorithm 1 and Algorithm 2

Threshold without gapProof of Theorem 1

17

After Algorithm 1, 2, we have :𝓝 𝑋 𝑖 𝑋 𝑖+1

min()

Next, find max() :

𝑥𝑖 𝑃= {𝑥𝑖+𝑢 }↑𝑑−𝑢

𝒖Apply a binary search algorithm to where each group test iscomposed of consecutive items.

18

Algorithm 1 FIND-TWO-CANDIDATES

𝑋 1 𝑋 3𝑋 2 𝑋 ⌈ 𝑛/𝑢⌉𝑋 4

𝓝𝑋 5 𝑋 6 ⋯⋯⋯⋯

Positive :

Negative :

Threshold without gap

19

Lemma 1 (Chang and Tsai, 2014)

FIND-TWO-CANDIDATES returns that

min in

tests.

Proof of theorem 1

20

Algorithm 2 LOCATE-STARTER𝑋 𝑖 𝑋 𝑖+1

𝒖

𝑋 𝑖 𝑋 𝑖+1

𝒖

𝑋 𝑖 𝑋 𝑖+1

𝒖

+¿ −

20

Algorithm 2 LOCATE-STARTER𝑋 𝑖 𝑋 𝑖+1

𝒖

𝑋 𝑖 𝑋 𝑖+1

𝒖

𝑋 𝑖 𝑋 𝑖+1

𝒖

𝑋 𝑖 𝑋 𝑖+1

𝒖

𝑋 𝑖 𝑋 𝑖+1

𝒖

+¿ −

+¿ −

Threshold without gap

21

Lemma 2 (Chang and Tsai, 2014)

LOCATE-STARTER can identify min() from

in

tests.

Proof of theorem 1

Threshold without gap

22

Theorem 1 (Chang and Tsai, 2014)

For gap-free T.G.T.C, all positives can be identified in

tests.

Threshold with gap

23

Theorem 2 (Chang and Tsai, 2014)

For T.G.T.C with , a -consecutive-approximate

set can be identified in

tests.

Main result

Sequential algorithm

Nonadaptive algorithm

25

Nonadaptive algo. for T.G.T.CRecall :

[1 1 0 1 0 1 0 0 00 1 0 0 1 1 0 1 01 0 1 0 0 1 1 1 01 0 0 1 0 0 1 0 10 1 1 0 1 0 0 1 01 1 0 0 1 0 1 1 0

]𝑐1𝑐2𝑐3𝑐4𝑐5𝑐6𝑐7𝑐8𝑐9

𝑡1𝑡 2𝑡 3𝑡 4𝑡5𝑡 6

100100

26

Consecutive-disjunct matrixDefinition 1 (Chang, Chiu and Tsai, 2014)

A binary matrix is -consecutive-disjunct if for any cyclically consecutive columns and other cyclically consecutive columns , there exists one row intersecting but none of .

the minimum number of rows among of all -consecutive-disjunct matrices of columns.

𝒘 𝒓

1111111 000000000

27

Consecutive-disjunct matrix

• Probabilistic method Lovsz Local Lemma (1974)

• Greedy construction Lovsz-Stein Theorem (1975)

28

Probabilistic method

Lemma 3

1. event.

2. For each is dependent of at most events.

3. for all .

If ,

then .

(Lovsz Local Lemma)

29

Probabilistic method

Theorem 3 (Chang, Chiu and Tsai, 2014)

with and ,

Example.

𝑡 (𝑛 , 2 ,1 ] ≤ 274

ln (8𝑛−24 )+ 274

.

30

Greedy construction

with and

where

Theorem 7 (Chang, Chiu and Tsai, 2014)

Example.

𝑡 (𝑛 , 2 ,1 ]<9𝑒2¿

𝑡 (𝑛 , 2 , 2 ]<16𝑒2¿

Nonadaptive algo. for T.G.T.C

31

Goal : Identify a -approximate set.

⋯

𝑑 𝑑 𝑑 𝑑 𝑑 𝑑 𝑑

32

Nonadaptive algo. for T.G.T.C

Given

[1 1 0 1 1 0 1 1 00 1 1 0 1 1 0 1 10 0 1 1 0 1 1 0 1 ]

Apply a

-consecutive-disjunct matrix

with columns.

33

Nonadaptive algo. for T.G.T.C

For T.G.T.C with , nonadaptive algorithm can identify

a -approximate set in tests.

Furthermore, the decoding complexity is .

Theorem 8 (Chang, Chiu and Tsai, 2014)

Proof.

.

34

Nonadaptive algo. for T.G.T.C

For G.T.C, nonadaptive algorithm can identify all positives in

tests. Furthermore, the decoding complexity is .

Theorem 9 (Chang, Chiu and Tsai, 2014)

Proof.

.

Concluding

35

Threshold group testing with consecutive positives

• Lower bound

.

• Sequential algorithm

.

• Nonadaptive algorithm

and

decoding complexity : .

References

1. D. J. Balding and D. C. Torney, The design of pooling experimentsfor screening a clone map, Fungal Genet. Biol. 21 (1997) 302-307.

2. H. Chang, Y.-C Chiu and Y.-L Tsai, A variation of cover-free families and its applications, preprint.

3. H. Chang and Y.-L Tsai, Threshold group testing with consecutivepositives, Discrete Appl. Math. 169 (2014) 68-72.

4. C. J. Colbourn, Group testing for consecutive positives, Ann. Combin. 3 (1999) 37-41.

5. P. Damaschke, Threshold group testing, In: General Theory ofInformation Transfer and Combinatorics, Lect. Notes Comput. Sci. 4123 (2006) 707-718.

6. P. Erdos and L. Lovasz, Infinite and finite sets, Colloq. Math. Soc. Janos Bolyai 10 (1974) 609-627.

7. J. S.-T. Juan and G. J. Chang, Adaptive group testing for consecutivepositives, Discrete Math. 308 (2008) 1124-1129.

36

References

8. L. Lovasz, On the ratio of optimal integral and fractional covers,Discrete Math. 13 (1975) 383-390.

9. R. A. Moser and G. Tardos, A constructive proof of the general Lovasz Local Lemma, Journal of the ACM (JACM). 57 (2010) 1-15.

10. M. Muller and M. Jimbo, Consecutive positive detectable matricesand group testing for consecutive positives, Discrete Math. 279(2004) 369-381.

11. S. K. Stein, Two combinatorial covering problems, J. CombinatorialTheory. 16 (1974) 391-397.

37

Thank you for your attention!