Advisor : Huilan Chang
Student : Yi-Lin Tsai
Department of Applied Mathematics
National University of Kaohsiung
2014/08/02
Outline
Introduction

•
Group testing
•
Group testing with consecutive positives
•
Threshold group testing
Main result
•
Sequential algorithm for T.G.T.C
•
Nonadaptive algorithm for T.G.T.C
Concluding
Reference
2
Classical group testing

• Given a set 𝒩 of 𝑛 items, each is either positive
or negative, and a set 𝒟 of at most 𝑑 positives.
• Goal : identify all positives by group test.
• Group Test : a test on a subset S ⊆ 𝒩.
positive
negative
Positive outcome:
S contains at least one
positive item.
3
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.
items
𝑐1 𝑐2 𝑐3 𝑐4 𝑐5 𝑐6 𝑐7 𝑐8 𝑐9
𝑡1 1 1 0 1 0 1 0 0 0
𝑡2 0 1 0 0 1 1 0 1 0
𝑡3 1 0 1 0 0 1 1 1 0
𝑡4 1 0 0 1 0 0 1 0 1
tests
𝑡5
𝑡6
0 1 1 0 1 0 0 1 0
1 1 0 0 1 0 1 1 0
4
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.
items
𝑐1 𝑐2 𝑐3 𝑐4 𝑐5 𝑐6 𝑐7 𝑐8 𝑐9
𝑡1 1 1 0 1 0 1 0 0 0 1
𝑡2 0 1 0 0 1 1 0 1 0 0
Outcome
𝑡3 1 0 1 0 0 1 1 1 0 0
vector
𝑡4 1 0 0 1 0 0 1 0 1 1
𝑡5 0 1 1 0 1 0 0 1 0 0
tests 𝑡6 1 1 0 0 1 0 1 1 0 0
4
Consecutive model

• 𝒩 = 𝑎1 , 𝑎2 , 𝑎3 , … 𝑎𝑛 is a set of items
with the linear order 𝑎𝑖 ≺ 𝑎𝑖+1 for 1 ≤ 𝑖 < 𝑛.
• 𝒟 : is a set of positive items which is
consecutive (under the ordering ≺),
and contains at most 𝒅 items.
• Test : choose arbitrary subset 𝑆 of 𝒩.
5
Consecutive model

• Balding and Torney (1997) and Colbourn (1999) first studied this model.
• Colbourn (1999)
sequential :
nonadaptive :
𝐥𝐨𝐠 𝟐 𝒏 + 𝐥𝐨𝐠 𝟐 𝒅 + 𝒄
𝐥𝐨𝐠 𝟐
𝒏
𝒅−𝟏
+ 𝟐𝒅 + 𝟏
• Muller and Jimbo (2004)
nonadaptive :
𝐥𝐨𝐠 𝟐
• Juan and Chang (2008)
sequential :
𝒏
𝒅−𝟏
+ 𝟐𝒅 − 𝟏
Lower bound :
𝐥𝐨𝐠 𝟐 (𝒅𝒏 + 𝟏 − 𝒅(𝒅 − 𝟏)/𝟐)
𝐥𝐨𝐠 𝟐 𝒏𝒅 + 𝟏, for 𝑛 ≥ 𝑑 − 1
6
Threshold group testing
• Peter Damaschke (2006)

arbitrary answer
negative
𝒍
lower threshold
positive
𝒖
upper threshold
7
Threshold group testing
• Peter Damaschke (2006)

𝒈
𝒍
lower threshold
𝒖
upper threshold
• 𝑔 =𝑢−𝑙−1
• If 𝑔 = 0 then we can find all positives.
If 𝑔 > 0 then we can only find a 𝑔-approximate set.
7
Threshold group testing
•

A set 𝑃 is called 𝑔-approximate
if 𝒟\𝑃 ≤ 𝑔 and 𝑃\𝒟 ≤ 𝑔.
EX1
Let 𝑢 = 3, 𝑙 = 1 ⇒ 𝑔 = 1.
a
c
b
d
𝟏-approximate set
e
EX2 The classical group testing is the case of
𝑢 = 1, 𝑙 = 0.
8
Our work
Group testing with
consecutive
positives
Threshold group
testing
Threshold group testing with
consecutive positives
9
Our work

Threshold group testing with consecutive positives
•
Lower bound
𝐥𝐨𝐠 𝟐 𝒏(𝒅 − 𝒖 + 𝟏) − 𝟏, 𝑛 ≥ 𝑑 + 𝑢 − 2.
•
Sequential algorithm
𝐥𝐨𝐠 𝟐 𝒏/𝒖 − 𝟏 + 𝟐 𝐥𝐨𝐠 𝟐 (𝒖 + 𝟐) + 𝐥𝐨𝐠 𝟐 (𝒅 − 𝒍) − 𝟐.
•
Nonadaptive algorithm
𝟏𝟓 𝐥𝐨𝐠 𝟐 𝒏𝒅 + 𝟒𝒅 + 𝟕𝟏 and
decoding complexity : 𝐎(𝒏𝒅 𝒍𝒐𝒈𝟐 𝒏𝒅 + 𝒖𝒅𝟐 ).
10
Sequential algorithm
Nonadaptive algorithm
Sequential algo. for T.G.T.C
Recall :
at most 𝑑
positives
…
It is usually assumed that 𝒖 ≤ 𝓓 ≤ 𝒅.
12
Sequential algo. for T.G.T.C
Information-theoretic lower bound :
Proposition (Chang and Tsai, 2014)
If 𝑛 ≥ 𝑑 + 𝑢 − 2, then the number of group tests
required to identify all positive items from 𝒩 is
at least
𝐥𝐨𝐠 𝟐 𝒏(𝒅 − 𝒖 + 𝟏) − 𝟏.
13
Sequential algo. for T.G.T.C
Our job :
Provide an algorithm to locate all positive items from linear
order 𝒩 and compare with the lower bound.
at most 𝑑
positives
…
14
Sequential algo. for T.G.T.C
Our job :
Provide an algorithm to locate all positive items from linear
order 𝒩 and compare with the lower bound.
at most 𝑑
positives
…
14
Sequential algo. for T.G.T.C
Our job :
Provide an algorithm to locate all positive items from linear
order 𝒩 and compare with the lower bound.
at most 𝑑
positives
…
min(𝒟)
max(𝒟)
≥ 𝒖 → positive.
We start with the case gap 𝑔 = 0.
< 𝒖 → negative.
14
Threshold without gap
Theorem 1 (Chang and Tsai, 2014)
For gap-free T.G.T.C, all positives can be identified in
𝐥𝐨𝐠 𝟐 𝒏/𝒖 − 𝟏 + 𝟐 𝐥𝐨𝐠 𝟐 (𝒖 + 𝟐)
+ 𝐥𝐨𝐠 𝟐 (𝒅 − 𝒖 + 𝟏) − 𝟐
tests.
15
Threshold without gap
Proof of Theorem 1
• First partition 𝒩 into 𝑛/𝑢 parts of 𝒖 consecutive items and
add some dummy negative items to the last part.
dummy
items
𝓝
𝑋1
𝑋2
𝑋3
𝑋4
⋯⋯⋯⋯
𝑋 𝑛/𝑢
⋯⋯⋯⋯
𝑋 𝑛/𝑢
• Let 𝒳 = 𝑋1 , 𝑋2 , ⋯ , 𝑋 𝑛/𝑢 .
• Goal : find min(𝒟).
𝓝
𝑋1
𝑋2
𝑋3
𝑋4
𝑋5
Algorithm 1 and Algorithm 2
16
Threshold without gap
Proof of Theorem 1
After Algorithm 1, 2, we have :
𝑋𝑖
𝑋𝑖+1
𝓝
min(𝒟)
Next, find max(𝒟) :
𝑥𝑖
𝑃 = 𝑥𝑖+𝑢 ↑𝑑−𝑢
𝒖
Apply a binary search algorithm to 𝑃 where each group test is
composed of 𝑢 consecutive items.
17
Algorithm 1 FIND-TWO-CANDIDATES
𝓝
𝑋1
𝑋2
𝑋3
𝑋4
𝑋5
𝑋6
⋯⋯⋯⋯
𝑋 𝑛/𝑢
Positive :
Negative :
18
Threshold without gap
Proof of theorem 1
Lemma 1 (Chang and Tsai, 2014)
FIND-TWO-CANDIDATES returns 𝑋𝑖 , 𝑋𝑖+1 that
min 𝒟 ∈ 𝑋𝑖 ∪ 𝑋𝑖+1 in
𝐥𝐨𝐠 𝟐 𝒏/𝒖 − 𝟏
tests.
19
Algorithm 2 LOCATE-STARTER
𝑋𝑖
𝑋𝑖+1
𝒖
+
𝑋𝑖
𝑋𝑖+1
𝒖
−
𝑋𝑖
𝑋𝑖+1
𝒖
20
Algorithm 2 LOCATE-STARTER
𝑋𝑖
𝑋𝑖+1
𝒖
−
+
𝑋𝑖
𝑋𝑖+1
𝑋𝑖
𝑋𝑖+1
𝒖
𝒖
+
𝑋𝑖
𝑋𝑖+1
𝒖
−
𝑋𝑖
𝑋𝑖+1
𝒖
20
Threshold without gap
Proof of theorem 1
Lemma 2 (Chang and Tsai, 2014)
LOCATE-STARTER can identify min(𝒟) from 𝑋𝑖 ∪ 𝑋𝑖+1
in
𝟐 𝐥𝐨𝐠 𝟐 (𝒖 + 𝟐) − 𝟐
tests.
21
Threshold without gap
Theorem 1 (Chang and Tsai, 2014)
For gap-free T.G.T.C, all positives can be identified in
𝐥𝐨𝐠 𝟐 𝒏/𝒖 − 𝟏 + 𝟐 𝐥𝐨𝐠 𝟐 (𝒖 + 𝟐)
+ 𝐥𝐨𝐠 𝟐 (𝒅 − 𝒖 + 𝟏) − 𝟐
tests.
22
Threshold with gap
Theorem 2 (Chang and Tsai, 2014)
For T.G.T.C with 𝑔 > 0, a 𝑔-consecutive-approximate
set can be identified in
𝐥𝐨𝐠 𝟐 𝒏/𝒖 − 𝟏 + 𝟐 𝐥𝐨𝐠 𝟐 (𝒖 + 𝟐)
+ 𝐥𝐨𝐠 𝟐 (𝒅 − 𝒍) − 𝟐
tests.
23
Sequential algorithm
Nonadaptive algorithm
Nonadaptive algo. for T.G.T.C
Recall :
𝑡1
𝑡2
𝑡3
𝑡4
𝑡5
𝑡6
𝑐1
1
0
1
1
0
1
𝑐2
1
1
0
0
1
1
𝑐3
0
0
1
0
1
0
𝑐4
1
0
0
1
0
0
𝑐5
0
1
0
0
1
1
𝑐6
1
1
1
0
0
0
𝑐7
0
0
1
1
0
1
𝑐8
0
1
1
0
1
1
𝑐9
0
0
0
1
0
0
1
0
0
1
0
0
25
Consecutive-disjunct matrix
Definition 1 (Chang, Chiu and Tsai, 2014)
A binary matrix is (𝒓, 𝒘]-consecutive-disjunct if for any 𝑤
cyclically consecutive columns 𝐶1 , ⋯ , 𝐶𝑤 and other 𝑟 cyclically
consecutive columns 𝐶𝑤+1 , ⋯ , 𝐶𝑤+𝑟 , there exists one row
intersecting 𝐶1 , ⋯ , 𝐶𝑤 but none of 𝐶𝑤+1 , ⋯ , 𝐶𝑤+𝑟 .
𝒘
𝒓
1111111
000000000
𝒕 𝒏, 𝒓, 𝒘 ≔ the minimum number of rows among of all
(𝑟, 𝑤]-consecutive-disjunct matrices of 𝑛 columns.
26
Consecutive-disjunct matrix
• Probabilistic method
Lovasz Local Lemma (1974)
• Greedy construction
Lovasz-Stein Theorem (1975)
27
Probabilistic method
Lemma 3 (Lovasz Local Lemma)
1.
𝐴𝑖 ∶ event.
2.
For each 𝐴𝑖 ∶ is dependent of at most 𝜇 events.
3.
𝑃𝑟(𝐴𝑖 ) ≤ 𝑝 for all 1 ≤ 𝑖 ≤ 𝑛.
If 𝑒𝑝 𝜇 + 1 ≤ 1,
then 𝑃𝑟 𝐴1 , 𝐴2 , ⋯ , 𝐴𝑛 > 0.
28
Probabilistic method
Theorem 3 (Chang, Chiu and Tsai, 2014)
∀ 𝑟, 𝑤, 𝑛 > 0 with 𝑛 ≥ 3𝑟 + 2𝑤 − 1 and 𝑟 ≥ 𝑤,
𝑡 𝑛, 𝑟, 𝑤 ≤
2 + 𝑤2
4𝑟
+
4𝑤
−
4
𝑛
−
6
𝑟
1 𝑟
𝑤
(𝑟 + 1) 1 + 𝑟 ln
−13𝑤𝑟 + 12𝑟 + 13𝑤 − 5
+1 .
Example.
27
𝑡 𝑛, 2, 1 ≤ 27
ln
8𝑛
−
24
+
4
4.
81
𝑡 𝑛, 2, 2 ≤ 81
ln
12𝑛
−
55
+
.
4
4
29
Greedy construction
Theorem 7 (Chang, Chiu and Tsai, 2014)
∀ 𝑟, 𝑛 > 0 with 𝑤 ≥ 1 and 𝑛 ≥ 𝑤 2 𝑘,
𝑡 𝑛, 𝑟, 𝑤 < 𝑒 2 𝑘 𝑤 ln
𝑘−1 2 𝑤 𝑘−2
𝑛 − 𝑘
𝑛+ 𝑟−1 𝑤−1
𝑘2
+1 ,
where 𝑘 = 𝑟 + 𝑤.
Example.
𝑡 𝑛, 2, 1 < 9𝑒 2 ln((2/9)𝑛2 − (1/3) 𝑛 + 1)
𝑡 𝑛, 2, 2 < 16𝑒 2 ln((3/16)𝑛2 − 𝑛 + 1 + 1)
30
Nonadaptive algo. for T.G.T.C
Goal : Identify a 𝑔-approximate set.
𝑑
𝑑
𝑑
𝑑
𝑑
𝑑
𝑑
⋯
31
Nonadaptive algo. for T.G.T.C
Apply a
(2,2]-consecutive-disjunct matrix
with
𝑛
𝑑
columns.
Given
1 1
0 1
0 0
𝑃
0
1
1
= 3 𝑎𝑛𝑑 𝑢
1 1 0
0 1 1
1 0 1
=2
1 1 0
0 1 1
1 0 1
32
Nonadaptive algo. for T.G.T.C
Theorem 8 (Chang, Chiu and Tsai, 2014)
For T.G.T.C with 𝑢 > 1, nonadaptive algorithm can identify
a 𝑔-approximate set in 𝟏𝟓 𝐥𝐨𝐠 𝟐 𝒏𝒅 + 𝟒𝒅 + 𝟕𝟏 tests.
Furthermore, the decoding complexity is 𝑶(𝒏𝒅 𝐥𝐨𝐠𝟐 𝒏𝒅 + 𝒖𝒅𝟐 ).
Proof.
𝑡
𝑛
𝑛
81
, 2, 2 + 4𝑑 ≤ 4 ln 12
− 55 + 81
+ 4𝑑
4
𝑑
𝑑
𝑛
≤ 15 log 2 𝑑 + 4𝑑 + 71 .
33
Nonadaptive algo. for T.G.T.C
Theorem 9 (Chang, Chiu and Tsai, 2014)
For G.T.C, nonadaptive algorithm can identify all positives in
𝟓 𝐥𝐨𝐠 𝟐 𝒏𝒅 + 𝟐𝒅 + 𝟐𝟏 tests. Furthermore, the decoding complexity
is 𝑶(𝒏𝒅 𝐥𝐨𝐠𝟐 𝒏𝒅 + 𝒅𝟐 ).
Proof.
𝑡
𝑛
𝑛
27
, 2, 1 + 2𝑑 ≤ 4 ln 8
− 24 + 27
+ 2𝑑
4
𝑑
𝑑
𝑛
≤ 5 log 2 𝑑 + 2𝑑 + 21 .
34
Concluding

Threshold group testing with consecutive positives
•
Lower bound
𝐥𝐨𝐠 𝟐 𝒏(𝒅 − 𝒖 + 𝟏) − 𝟏, 𝑛 ≥ 𝑑 + 𝑢 − 2.
•
Sequential algorithm
𝐥𝐨𝐠 𝟐 𝒏/𝒖 − 𝟏 + 𝟐 𝐥𝐨𝐠 𝟐 (𝒖 + 𝟐) + 𝐥𝐨𝐠 𝟐 (𝒅 − 𝒍) − 𝟐.
•
Nonadaptive algorithm
𝟏𝟓 𝐥𝐨𝐠 𝟐 𝒏𝒅 + 𝟒𝒅 + 𝟕𝟏 and
decoding complexity : 𝐎(𝒏𝒅 𝐥𝐨𝐠𝟐 𝒏𝒅 + 𝒖𝒅𝟐 ).
35
References

1. D. J. Balding and D. C. Torney, The design of pooling experiments
for 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 consecutive
positives, 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 of
Information 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 consecutive
positives, 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 matrices
and group testing for consecutive positives, Discrete Math. 279
(2004) 369-381.
11. S. K. Stein, Two combinatorial covering problems, J. Combinatorial
Theory. 16 (1974) 391-397.
37