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 .
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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
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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
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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
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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.
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Algorithm 1 FIND-TWO-CANDIDATES
𝑋 1 𝑋 3𝑋 2 𝑋 ⌈ 𝑛/𝑢⌉𝑋 4
𝓝𝑋 5 𝑋 6 ⋯⋯⋯⋯
Positive :
Negative :
Threshold without gap
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Lemma 1 (Chang and Tsai, 2014)
FIND-TWO-CANDIDATES returns that
min in
tests.
Proof of theorem 1
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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
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Lemma 2 (Chang and Tsai, 2014)
LOCATE-STARTER can identify min() from
in
tests.
Proof of theorem 1
Threshold without gap
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Theorem 1 (Chang and Tsai, 2014)
For gap-free T.G.T.C, all positives can be identified in
tests.
Threshold with gap
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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
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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
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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
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Consecutive-disjunct matrix
• Probabilistic method Lovsz Local Lemma (1974)
• Greedy construction Lovsz-Stein Theorem (1975)
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Probabilistic method
Lemma 3
1. event.
2. For each is dependent of at most events.
3. for all .
If ,
then .
(Lovsz Local Lemma)
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Probabilistic method
Theorem 3 (Chang, Chiu and Tsai, 2014)
with and ,
Example.
𝑡 (𝑛 , 2 ,1 ] ≤ 274
ln (8𝑛−24 )+ 274
.
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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
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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.
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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.
.
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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
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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.
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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.
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Thank you for your attention!