2014/08/02
Error-correcting Pooling Designs
and
Group Testing for Consecutive Positives
Advisor：Huilan Chang
Student：Yi-Tsz Tsai
Department of Applied Mathematics
National Kaohsiung University
2014/08/02
Outline


Error-correcting pooling designs

Constructed from vectors

Constructed from distance-regular graph
Two-stage algorithm for Group testing
of consecutive positives
1
2014/08/02
Pooling designs
Classical group testing
𝑛
items where each item is either positive or
negative.
 Information：at most 𝑑 positives. (𝑑 ≪ 𝑛)
 Goal：identify all positives by group tests.
Positive
Negative
2
2014/08/02
Pooling designs
 Types


of group testing algorithm：
Sequential algorithm：
 tests are conducted one by one.
Nonadaptive algorithm (Pooling design)：
 all tests (pools) are designed simultaneously.
 find all positives from the testing outcomes.
3
2014/08/02
Pooling designs
Nonadaptive algorithm
 Binary



matrix representation：
Rows are tests.
Columns are items.
An entry 𝑎𝑖𝑗 = 1 if test 𝑖 contains item 𝑗.
Testing
outcome
𝑡1
𝑡2
𝑡3
𝑡4
1
0
0
1
0
1
0
0
0
0
1
1
1
1
1
0
0
1
1
0
0
1
0
1
1
0
1
1
4
2014/08/02
Pooling designs
Nonadaptive algorithm

A binary matrix 𝑀 is 𝒔-disjunct if any 𝑠 + 1 columns
of 𝑀 with one designated, there is a row intersecting
the designated column and none of the other 𝑠 columns.
𝐶0
𝑠 columns
⋮
At least 1 row
⋯
⋯
1
0
⋯
0
⋯
⋮
⋮
5
2014/08/02
Pooling designs
Nonadaptive algorithm

A binary matrix 𝑀 is 𝒔𝒆 -disjunct if any 𝑠 + 1 columns
of 𝑀 with one designated, there are 𝑒 + 1 rows
intersecting the designated column and none of the other
𝑠 columns
𝐶
𝑠 columns
0
At least 𝑒 + 1 rows
⋯
⋯
1
1
1
0
0
0
1
1
0
0
⋮
⋮
⋮
⋯
⋯
⋯
0
0
0
⋯
⋯
0
0
⋯
6
2014/08/02
Pooling designs
Nonadaptive algorithm
 Note：
0
 An 𝑠 -disjunct


matrix is also called 𝑠-disjunct.
An 𝑠 𝑒 -disjunct matrix is fully 𝒔𝒆 -disjunct
if it is not 𝑠1 𝑒1 -disjunct whenever 𝑠1 > 𝑠 or
𝑒1 > 𝑒.
Application：
A 𝑑 𝑒 -disjunct matrix is 𝑒 2 -error-correcting.
7
2014/08/02
Construction
Error-correcting pooling designs


Constructed from vectors
Constructed from distance-regular graph
8
2014/08/02
Construction - sequences
Error-correcting pooling designs

Constructed from vectors



A 𝑞-ary vector is a vector whose entries are from
0,1,2, … , 𝑞 .
The weight of a vector ≔ # of its nonzero entries.
[𝑛]
𝑘
[𝑞]𝑘 ≔ all 𝑞-ary vectors of length 𝑛 and weight 𝑘.
9
2014/08/02
Construction - sequences
Definition (D’ychakov et al., 05’)
Let 1 ≤ 𝑑 ≤ 𝑘 ≤ 𝑛 and 𝑞 ≥ 1.
𝝅𝒒 (𝒅, 𝒌, 𝒏)：binary matrix by rows indexed
and columns indexed by
For 𝛼 ∈
𝛽.



[𝑛]
𝑑
[𝑞]𝑑 and 𝛽
[𝑛]
𝑘 .
[𝑞]
𝑘
𝑘
∈ [𝑛]
[𝑞]
,
𝑘
[𝑛]
𝑑
[𝑞]𝑑
𝜋 𝛼, 𝛽 = 1 iff 𝛼 ≺
𝛽(𝑖)：the 𝑖-th entry of 𝛽.
α ≺ 𝛽：if for 1 ≤ 𝑖 ≤ 𝑛, 𝛼 𝑖 = 𝛽(𝑖) whenever 𝛼 𝑖 ≠ 0.
Example：In 𝜋2 (3,4,6) , (1,2,0,0,1,0) ≺ (1,2,2,0,1,0)
(1,2,2,0,1,0)
(1,2,0,0,1,0)
1
10
2014/08/02
Construction - sequences
Theorem (D’ychakov et al., 05’)
Let 1 ≤ 𝑠 ≤ 𝑑 ≤ 𝑘 ≤ 𝑛 and 𝑞 ≥ 1.
𝜋𝑞 (𝑑, 𝑘, 𝑛) is fully 𝑠 𝑒 -disjunct where 𝑒 =
𝑘−𝑠
𝑑−𝑠
− 1.
How about “𝛼 𝑖 = 𝛽(𝑖) for some 𝛼 𝑖 ≠ 0”？
11
2014/08/02
Construction - sequences
Definition 1
Let 1 ≤ 𝑑 ≤ 𝑘 ≤ 𝑛 and 𝑞 ≥ 1.
𝝅𝒒 (𝒓; 𝒅, 𝒌, 𝒏)：binary matrix by rows indexed
and columns indexed by
𝑑
For 𝛼 ∈ [𝑛]
[𝑞]
and 𝛽
𝑑
𝛼 ∩ 𝛽 = 𝑟.
[𝑛]
𝑘
[𝑞]
.
𝑘
𝑘
∈ [𝑛]
[𝑞]
,
𝑘
[𝑛]
𝑑
[𝑞]𝑑
𝜋 𝛼, 𝛽 = 1 iff

𝛼 ∩ 𝛽 ≔ 𝛾, where 𝛾 is a 𝑞−ary vector of lenth 𝑛 with
𝛾 𝑖 = 𝛼(𝑖) if 𝛼 𝑖 = 𝛽(𝑖) ≠ 0 and 𝛾 𝑖 = 0 for otherwise.

Example：
α ∩ 𝛽 = 1,0,1,2,0 ∩ 1,3,2,2,0
12
2014/08/02
Construction - sequences
Definition 1
Let 1 ≤ 𝑑 ≤ 𝑘 ≤ 𝑛 and 𝑞 ≥ 1.
𝝅𝒒 (𝒓; 𝒅, 𝒌, 𝒏)：binary matrix by rows indexed
and columns indexed by
𝑑
For 𝛼 ∈ [𝑛]
[𝑞]
and 𝛽
𝑑
𝛼 ∩ 𝛽 = 𝑟.
[𝑛]
𝑘
[𝑞]
.
𝑘
𝑘
∈ [𝑛]
[𝑞]
,
𝑘
[𝑛]
𝑑
[𝑞]𝑑
𝜋 𝛼, 𝛽 = 1 iff

𝛼 ∩ 𝛽 ≔ 𝛾, where 𝛾 is a 𝑞−ary vector of lenth 𝑛 with
𝛾 𝑖 = 𝛼(𝑖) if 𝛼 𝑖 = 𝛽(𝑖) ≠ 0 and 𝛾 𝑖 = 0 for otherwise.

Example：
α ∩ 𝛽 = 1,0,1,2,0 ∩ 1,3,2,2,0 = (1,0,0,2,0)
⇒ 𝛼∩𝛽 =2
13
2014/08/02
Construction - sequences
 Example： 𝜋 𝛼, 𝛽 = 1 iff 𝛼 ∩ 𝛽 = 𝑟
𝜋2 (1; 2,3,4)
(1110)
(1001)
0
0
1
0
0
1
0
1
1
0
0
1
0
0
0
0
0
0
⋯⋯⋯
(0110)
(1011)
(0202)
(0022)
(1102)
(2022)
(1120)
1
1
1
0
(2022)
⋯⋯⋯
0
1
1
1
0
1
1
1
⋯⋯⋯
⋯⋯⋯
⋯⋯⋯
⋯⋯⋯
0
0
0
0
0
1
1
1
⋯⋯⋯
⋯⋯⋯
1
0
⋯⋯⋯
(1100)
(1010)
(0111)
0
0
0
0
0
0
⋯⋯⋯
(1101)
1
0
14
2014/08/02
Construction - sequences
Our result：
Theorem 1
𝑑 + 1 2 ≤ 𝑟 ≤ 𝑑 ≤ 𝑘 and 𝑛 − 𝑘 −
𝑠 𝑘 + 𝑑 − 2𝑟 ≥ 𝑑 − 𝑟.
Then 𝜋𝑞 (𝑟; 𝑑, 𝑘, 𝑛) is 𝑠 𝑒 -disjunct, where
𝑛−𝑘−𝑠−(𝑘+𝑑−2𝑟) 𝑑−𝑟
𝑒 = 𝑘−𝑠
𝑞
− 1.
𝑟−𝑠
𝑑−𝑟
Let 1 ≤ 𝑠 ≤ 𝑟,
Theorem 2
Let 1 ≤ 𝑠 ≤ 𝑟, 𝑑 + 1 2 ≤ 𝑟 ≤ 𝑑 ≤ 𝑘 ≤ 𝑛 and 𝑞 >
𝑠 + 1.
Then 𝜋𝑞 (𝑟; 𝑑, 𝑘, 𝑛) is 𝑠 𝑒 -disjunct, where
𝑛−𝑟
𝑑−𝑟 −1.
𝑒 = 𝑘−𝑠
(𝑞
−
𝑠
−
1)
𝑟−𝑠 𝑑−𝑟
15
2014/08/02
Construction - sequences
Analysis：
designated
𝑞-ary vector of weight 𝑑
Goal：
How many 𝛼
that 𝛼 ∩ 𝛽0 = 𝑟
and 𝛼 ∩ 𝛽𝑗 ≠ 𝑟
for each 𝑗𝜖[𝑠] ?
𝛼1
𝛼2
𝛼
𝑛
𝑑
𝑞-ary vector of weight 𝑘
𝛽0 𝛽1 ⋯ ⋯ 𝛽𝑠
1
1
1
0
0
0
1
1
0
0
⋮
⋯
⋯
⋯
0
0
0
⋯
⋯
0
0
⋯⋯ ⋯⋯ 𝛽 𝑛
𝑘
𝑞𝑘
𝑞𝑑
16
2014/08/02
Construction - sequences
Analysis： (Theorem 2)

Step1：Find the lower bound of the number of 𝑞-ary vectors 𝛾
of weight 𝑟 satisfying 𝛾 ≺ 𝛽0 and 𝛾 ⊀ 𝛽𝑗 for each 𝑗 ∈ 𝑠 .
𝑋0 ={𝑥1 , … , 𝑥𝑠 ：𝛽0 𝑥𝑗 ≠ 0 and
either 𝛽𝑗 𝑥𝑗 = 0 or 𝛽𝑗 (𝑥𝑗 ) ≠ 0
and 𝛽0 (𝑥𝑗 ) ≠ 𝛽𝑗 (𝑥𝑗 )}
…
…
…
)
𝛽0 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
…
…
)
…
…
…
…
)
…
…
…
…
)
…
…
…
…
)
𝛽1 (
≠
𝛽2 (
≠
𝛾 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
⋯
≠
𝛽𝑠 (
17
2014/08/02
Construction - sequences
Analysis： (Theorem 2)

Step1：Find the lower bound of the number of 𝑞-ary vectors 𝛾
of weight 𝑟 satisfying 𝛾 ≺ 𝛽0 and 𝛾 ⊀ 𝛽𝑗 for each 𝑗 ∈ 𝑠 .
𝑋0 ={𝑥1 , … , 𝑥𝑠 ：𝛽0 𝑥𝑗 ≠ 0 and
either 𝛽𝑗 𝑥𝑗 = 0 or 𝛽𝑗 (𝑥𝑗 ) ≠ 0
and 𝛽0 (𝑥𝑗 ) ≠ 𝛽𝑗 (𝑥𝑗 )}
𝑟 − 𝑋0
𝛾 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
𝛽0 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
)
≠…
0
…
…
)
…
…
…
…
)
…
…
…
…
)
…
…
…
…
)
=
…
≠
𝛽2 (
≠
𝛽1 (
⋯
≠
𝛽𝑠 (
17
2014/08/02
Construction - sequences
Analysis： (Theorem 2)

Step1：Find the lower bound of the number of 𝑞-ary vectors 𝛾
of weight 𝑟 satisfying 𝛾 ≺ 𝛽0 and 𝛾 ⊀ 𝛽𝑗 for each 𝑗 ∈ 𝑠 .
𝑋0 ={𝑥1 , … , 𝑥𝑠 ：𝛽0 𝑥𝑗 ≠ 0 and
either 𝛽𝑗 𝑥𝑗 = 0 or 𝛽𝑗 (𝑥𝑗 ) ≠ 0
and 𝛽0 (𝑥𝑗 ) ≠ 𝛽𝑗 (𝑥𝑗 )}
𝑟 − 𝑋0
𝑘 − 𝑋0
𝑟 − 𝑋0
𝑘−𝑠
≥
𝑟−𝑠
𝛾 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
𝛽0 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
≠…
0
…
…
)
…
…
…
…
)
…
…
…
…
)
…
…
…
…
)
0 0 … 0 0 … 0 )
=
≠
𝛽2 (
≠
𝛽1 (
⋯
≠
𝛽𝑠 (
17
2014/08/02
Construction - sequences
Analysis： (Theorem 2)

Step2：Find
•
?
𝑑−𝑟
Fixed 𝛾, choose 𝑖 satisfying 𝛼 𝑖 ∈ 𝑞 ∖ { 𝛽𝑗 𝑖 ≠ 0：𝑗 = 0,1, … , 𝑠}
wherever 𝛾 𝑖 = 0
…
𝛼 (
…
…
…
)
𝛾 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
𝛽0 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
≠…
0
…
…
)
…
…
…
…
)
…
…
…
…
)
…
…
…
…
)
0 0 … 0 0 … 0 )
=
≠
𝛽2 (
≠
𝛽1 (
⋯
≠
𝛽𝑠 (
18
2014/08/02
Construction - sequences
Analysis： (Theorem 2)

?
Step2：Find
•
𝑑−𝑟
Fixed 𝛾, choose 𝑖 satisfying 𝛼 𝑖 ∈ 𝑞 ∖ { 𝛽𝑗 𝑖 ≠ 0：𝑗 = 0,1, … , 𝑠}
wherever 𝛾 𝑖 = 0
(𝑑 − 𝑟)
…
𝛼 (
…
𝛽0 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
≠…
0
=
≠
𝛽2 (
≠
𝛽1 (
…
…
)
0 0 … 0 0 … 0 )
≠
=
=
𝛾 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
…
…
)
…
…
…
)
…
…
…
…
)
…
…
…
…
)
⋯
…
≠
𝛽𝑠 (
18
2014/08/02
Construction - sequences
Analysis： (Theorem 2)

?
Step2：Find
•
𝑑−𝑟
Fixed 𝛾, choose 𝑖 satisfying 𝛼 𝑖 ∈ 𝑞 ∖ { 𝛽𝑗 𝑖 ≠ 0：𝑗 = 0,1, … , 𝑠}
wherever 𝛾 𝑖 = 0
𝑛−𝑟
𝑑−𝑟
(𝑑 − 𝑟)
…
𝛼 (
…
𝛽0 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
≠…
0
=
≠
𝛽2 (
≠
𝛽1 (
…
(𝑞 − 𝑠 − 1)
0 … 0 )
0 0 … 0 0 … 0 )
≠
=
=
𝛾 ( 𝑎1 𝑎2 𝑎3 … 𝑎|𝑋0|
…
𝑑−𝑟
…
…
)
…
…
…
)
…
…
…
…
)
…
…
…
…
)
⋯
…
≠
𝛽𝑠 (
18
2014/08/02
Construction - sequences
Concluding 1：
D’ychakov et al. (2005)：
by “containment relation”
(1) 𝑒1 =
𝑘−𝑠
𝑑−𝑠
−1
(2) 𝑒2 =
𝑘−𝑠
𝑟−𝑠
𝑛−𝑘−𝑠−(𝑘+𝑑−2𝑟)
𝑑−𝑟
𝑘−𝑠
𝑟−𝑠
𝑛−𝑟
𝑑−𝑟
(3) 𝑒2 =
𝑞 𝑑−𝑟 − 1
Our results：
by “intersecting relation”
(𝑞 − 𝑠 − 1)𝑑−𝑟 −1
𝑒2 +1
𝑛→∞ 𝑒1 +1
For given 𝑘 and 𝑑, if 𝑟 < 𝑑, then lim
= ∞,
where 𝑒2 = (2) or (3).
19
2014/08/02
Construction – Johnson graph
Error-correcting pooling designs

Constructed from distance-regular graph


The Johnson graph 𝑱(𝒏, 𝒕) is defined on [𝑛]
such that
𝑡
two vertices 𝑢 and 𝑣 are adjacent iff 𝑢 ∩ 𝑣 = 𝑡 − 1.
Binary matrix 𝑀 with columns and rows indexed by
𝑡-cliques.
20
2014/08/02
Construction – Johnson graph
 𝒕-clique：


For a connected graph 𝐺 = (𝑉, 𝐸)：
an 𝑙-subset 𝐶 of 𝑉 is a 𝑡-clique if any two vertices in
𝐶 are at distance 𝑡.
In Johnson graph 𝐽(𝑛, 𝑡)：
a 𝑡-clique with size 𝑙 is a collection of 𝑙 disjoint 𝑡subsets of 𝑛 .
 Example：𝐽(6,3)
456
123
124
125
• 𝑒 = 𝑢, 𝑣 if 𝑢 ∩ 𝑣 = 𝑡 − 1
245
236
235
234
21
2014/08/02
Construction – Johnson graph
 𝒕-clique：


For a connected graph 𝐺 = (𝑉, 𝐸)：
an 𝑙-subset 𝐶 of 𝑉 is a 𝑡-clique if any two vertices in
𝐶 are at distance 𝑡.
In Johnson graph 𝐽(𝑛, 𝑡)：
a 𝑡-clique with size 𝑙 is a collection of 𝑙 disjoint 𝑡subsets of 𝑛 .
 Example：𝐽(6,3)
456
123
124
125
• 𝑒 = 𝑢, 𝑣 if 𝑢 ∩ 𝑣 = 𝑡 − 1
•
123 , (456) is a 3-clique
of size 2.
245
236
235
234
21
2014/08/02
Construction – Johnson graph
Definition (Bai et al., 09’)
Let 𝑑 < 𝑘 and 𝑘𝑡 ≤ 𝑛.
𝑴(𝒏, 𝒕, 𝒅, 𝒌)：binary matrix with rows indexed by 𝑡clique with size 𝑑 and columns indexed by 𝑡-clique with
size 𝑘 such that 𝑀 𝐴, 𝐵 = 1 iff 𝐴 ⊆ 𝐵.
1,3
2,4
5,6
⋯⋯⋯
1,2
1
1
1
0
⋯⋯⋯
0
1,3
0
0
0
1
⋯⋯⋯
0
0
0
1
0
0
0
0
1
⋯⋯⋯
⋯⋯⋯
0
0
4,6
5,6
1,6
2,5
3,4
⋯
1,2
3,6
4,5
⋯
1,2
3,5
4,6
⋯
𝑀(6,2,1,3)
1,2
3,4
5,6
22
2014/08/02
Construction – Johnson graph
Theorem (Bai et al., 09’)
Let 1 ≤ 𝑠 ≤ 𝑑 < 𝑘 and 𝑘𝑡 ≤ 𝑛.
𝑀(𝑛, 𝑡, 𝑑, 𝑘) is fully 𝑠 𝑒 -disjunct where 𝑒 =
𝑘−𝑠
𝑑−𝑠
− 1.
How about “𝑀 𝐴, 𝐵 = 1 iff 𝐴 ∩ 𝐵 = 𝑖”？
23
2014/08/02
Construction – Johnson graph
Definition 2
Let 1 ≤ 𝑖 ≤ 𝑑 < 𝑘 and 𝑘𝑡 ≤ 𝑛.
𝑴(𝒊; 𝒏, 𝒕, 𝒅, 𝒌)：binary matrix with rows indexed by 𝑡clique with size 𝑑 and columns indexed by 𝑡-clique with
size 𝑘 such that
𝑀 𝐴, 𝐵 = 1 iff 𝐴 ∩ 𝐵 = 𝑖.
1,2
3,6
4,5
1,3
2,4
5,6
⋯⋯⋯
1,2
3,4
0
1
1
0
⋯⋯⋯
1
1,2
3,5
1
0
1
0
⋯⋯⋯
0
0
0
1
0
⋯⋯⋯
⋯
𝑀(1; 6,2,2,3)
3,6
4,5
1,6
2,5
3,4
⋯
1,2
3,5
4,6
⋯
1,2
3,4
5,6
0
24
2014/08/02
Construction – Johnson graph
Theorem (Lv et al., 14’)
Let 1 ≤ 𝑠 ≤ 𝑖, (𝑑 + 1) 2 ≤ 𝑖 ≤ 𝑑 < 𝑘 and
𝑛 − 𝑘𝑡 − 𝑠𝑡 𝑘 + 𝑑 − 2𝑖 ≥ 𝑑 − 𝑖 𝑡.
Then 𝑀(𝑖; 𝑛, 𝑡, 𝑑, 𝑘) is 𝑠 𝑒 -disjunct, where
𝑒=
𝑘−𝑠
𝑖−𝑠
𝑛−𝑘𝑡−𝑠𝑡(𝑘+𝑑−2𝑖)
𝑑−𝑖 𝑡
𝑑−𝑖 𝑡 !
𝑡! 𝑑−𝑖 𝑑−𝑖 !
− 1.
25
2014/08/02
Construction – Johnson graph
Our result：
Theorem 3
Let 1 ≤ 𝑠 ≤ 𝑖 ≤ 𝑑 − 1 < 𝑘 and 𝑘𝑡 ≤ 𝑛.
Then 𝑀(𝑖; 𝑛, 𝑡, 𝑑, 𝑘) is 𝑠 𝑒 -disjunct, where 𝑒 =
𝑘−𝑠
𝑖−𝑠
𝑛−𝑖𝑡
𝑑−𝑖 𝑡
𝑑−𝑖 𝑡 !
𝑡! 𝑑−𝑖 𝑑−𝑖 !
−(𝑘 + 𝑠 𝑘 + 𝑑 − 2𝑖 − 𝑖)
𝑛− 𝑖+1 𝑡
𝑑−𝑖−1 𝑡
𝑑−𝑖−1 𝑡 !
𝑡! 𝑑−𝑖−1 𝑑−𝑖−1 !
−1.
26
2014/08/02
Construction – Johnson graph
Concluding 2：

Lv et al. (14’) ：𝑒1 =

Our result：𝑒2 =
𝑘−𝑠
𝑖−𝑠
𝑘−𝑠
𝑖−𝑠
𝑛−𝑘𝑡−𝑠𝑡(𝑘+𝑑−2𝑖)
𝑑−𝑖 𝑡
𝑛−𝑖𝑡
𝑑−𝑖 𝑡
𝑒2 − 𝑒1 > Ω(𝑛
−
𝑡! 𝑑−𝑖 𝑑−𝑖 !
1
𝑑−𝑖 𝑡 !
𝑡! 𝑑−𝑖 𝑑−𝑖 !
−(𝑘 + 𝑠 𝑘 + 𝑑 − 2𝑖 − 𝑖)

𝑑−𝑖 𝑡 !
𝑛− 𝑖+1 𝑡
𝑑−𝑖−1 𝑡
𝑑−𝑖−1 𝑡 !
𝑡! 𝑑−𝑖−1 𝑑−𝑖−1 !
−1
𝑑−𝑖 𝑡−1 )
Table 1：Some comparisons of error-tolerance
capabilities of 𝑀(𝑖; 𝑛, 𝑡, 𝑑, 𝑘)
27
2014/08/02
Conclusion-Pooling designs
Our results for error-correcting capabilities：
 Construction


by intersecting relation：
𝑞-ary vectors

𝑒=
𝑘−𝑠
𝑟−𝑠
𝑛−𝑘−𝑠−(𝑘+𝑑−2𝑟)
𝑑−𝑟

𝑒=
𝑘−𝑠
𝑟−𝑠
𝑛−𝑟
𝑑−𝑟
𝑞𝑑−𝑟 − 1
(𝑞 − 𝑠 − 1)𝑑−𝑟 −1
𝑡-cliques of Johnson graph 𝐽(𝑛, 𝑡)

𝑒=
𝑘−𝑠
𝑟−𝑠
𝑛−𝑖𝑡
𝑑−𝑖 𝑡
𝑑−𝑖 𝑡 !
𝑡! 𝑑−𝑖 𝑑−𝑖 !
−(𝑘 + 𝑠 𝑘 + 𝑑 − 2𝑖 − 𝑖)
𝑛− 𝑖+1 𝑡
𝑑−𝑖−1 𝑡
𝑑−𝑖−1 𝑡 !
𝑡! 𝑑−𝑖−1 𝑑−𝑖−1 !
−1
28
2014/08/02
Two-stage for CGT
Two-stage algorithm for
group testing of consecutive positives
29
2014/08/02
Two-stage for CGT
Group testing of consecutive
positives
of objects 𝒩 ≔ 𝑣1 , 𝑣2 , … , 𝑣𝑛 satisfying the
linear order 𝑣𝑖 ≺ 𝑣𝑖+1 .
 A set
are consecutive under ≺.
Information：at most 𝑑 positives. (𝑑 ≪ 𝑛)
 Positives
 Motivation:
applications in DNA sequencing.
30
2014/08/02
Two-stage for CGT
Group testing of consecutive
positives

Nonadaptive algorithm：


Begin by partition 𝒩 into 𝑛𝑑 parts
Each part contains 𝑑 consecutive items.
All positive items are contained in at most two parts.
31
2014/08/02
Two-stage for CGT
Group testing of consecutive
positives

Nonadaptive algorithm：

Begin by partition 𝒩 into 𝑛𝑑 parts.
Each part contains 𝑑 consecutive items.
 All positive items are contained in at most two parts.
Colburn (99’) ：

Gray code ⇒ log 2

𝑛
𝑑−1
+ 2𝑑 + 1 tests.
Muller and Jimbo (04’)：
consecutive positive detectable matrices
𝑛
⇒ log 2 𝑑−1
+ 2𝑑 − 1 tests.
32
2014/08/02
Two-stage for CGT
Multi-stage algorithm
 Multi-stage
algorithm：Stages are sequential and
all tests in a stage are nonadaptive.
 Example：𝒩
= {𝑎, 𝑏, … , ℎ} and at most 2 positives.
called a “trivial two-stage algorithm”
if Stage 2 = identity matrix.
 Especially
33
2014/08/02
Two-stage for CGT
(𝒌, 𝒎, 𝒏)-selector ：
Definition (De Bonis et al., 05’)
Given 1 ≤ 𝑚 ≤ 𝑘 < 𝑛, a 𝑡 × 𝑛 binary matrix 𝑀 is a
(𝑘, 𝑚, 𝑛)-selector if any submatrix of 𝑀 obtained by
choosing 𝑘 out of 𝑛 arbitrary columns of 𝑀 contains at
least 𝑚 distinct rows of the identity matrix 𝐼𝑘 .
Arbitrary 5 columns
(5,3, 𝑛)-selector
⋯
At least 3 rows
of 𝐼5
1
0
0
0
0
0
1
0
0
1
0
0
⋮
⋮
0
1
0
1
0
1
0
0
⋯
34
2014/08/02
Two-stage for CGT
Trivial two-stage algorithm (De Bonis et al., 05’)：
 Partition
𝒩 into
𝑛
𝑑
parts ≔ 𝑋1 , 𝑋2 , … 𝑋 𝑛
Stage 1
𝑋1 𝑋2 𝑋3
(4,3,
𝑛
𝑑
…
𝑋𝑛
)-selector
𝑑
.
Stage 2
𝑋𝑘
𝑋𝑗
𝑋𝑖
𝑑
At most
3 parts left
Identity
Theorem (De Bonis et al., 05’)
Let 1 ≤ 𝑚 ≤ 𝑘 < 𝑛, there exists a (𝑘, 𝑚, 𝑛)-selector of
size 𝑡, with
𝑡<
𝑒𝑘 2
𝑛
ln
𝑘−𝑚+1
𝑘
+ 𝑒𝑘(2𝑘−1)
𝑘−𝑚+1
35
2014/08/02
Two-stage for CGT
Trivial two-stage algorithm (De Bonis et al., 05’) ：
Theorem 4
This trivial two-stage algorithm identifies all positives in
16 log 2 𝑛 𝑑 + 14𝑒 + 3𝑑 group tests.
Furthermore, its decoding complexity is


𝑛
𝑛
𝑂( log
𝑑
𝑑
+ 𝑑).
(𝑘, 𝑚, 𝑛)-selectors were not specifically introduced to
deal with the group testing of consecutive positives.
Next, we consider its variation
((𝑘1 , 𝑘2 ), 𝑚, 𝑛)-selectors
36
2014/08/02
Two-stage for CGT
((𝒌𝟏 , 𝒌𝟐 ), 𝒎, 𝒏)-selector ：
Definition 3
For 1 ≤ 𝑚 ≤ 𝑘1 + 𝑘2 ≤ 𝑛 and 𝑘1 + 𝑘2 = 𝑘,
A 𝑡 × 𝑛 binary matrix 𝑀 is a ((𝑘1 , 𝑘2 ), 𝑚, 𝑛)-selector if
any 𝑡 × (𝑘1 + 𝑘2 ) submatrix of 𝑀 obtained by choosing
𝑘1 consecutive columns and other 𝑘2 arbitrary columns
contains at least 𝑚 distinct rows of the identity matrix 𝐼𝑘 .




((2,2), 3, 𝑛)-selector
𝑘1 = 2
𝑘2 = 2
𝑚 = 3 rows of 𝐼4
(in the submatrix)
arbitrary
consecutive
⋯
1
1
0
0
0
0
1
0
0
1
0
1
0
1
0
0
⋯
37
2014/08/02
Two-stage for CGT
Our two-stage algorithm：
 Partition
 Stage
𝒩 into
𝑛
𝑑
parts ≔ 𝑋1 , 𝑋2 , … 𝑋 𝑛
𝑑
.
1：
𝑛
𝑑
Use a ((2,2), 3, )-selector as a pooling design
where the 𝑖-th column associated with 𝑋𝑖 .
 Discard each part contained in any negative test.
 Stage 2：Identity matrix.

Stage 1
𝑋1 𝑋2 𝑋3
((2,2), 3,
…
𝑛
𝑑
𝑋𝑛
Stage 2
𝑋𝑘
𝑋𝑗
𝑋𝑖
𝑑
At most
? parts left
)-selector
Identity
38
2014/08/02
Two-stage for CGT
Lemma
After using a ((2,2), 3, 𝑛 𝑑 )-selector, there remain at
most 3 parts in Stage 1 and their union contains all
positive items.

Note：
𝑡 𝑘1 , 𝑘2 , 𝑚, 𝑛 ≔ the min. number of rows among all
𝑘1 , 𝑘2 , 𝑚, 𝑛 -selectors.
Stage 2 ： Test each item in the remaining parts individually
⇒ There are 𝑡 2,2 , 3, 𝑛 𝑑

+ 3𝑑 tests.
Next, the upper bound of 𝑡 𝑘1 , 𝑘2 , 𝑚, 𝑛 ？
39
2014/08/02
Two-stage for CGT
Theorem 5 ( by Lovasz-Stein Theorem)
Let 1 ≤ 𝑚 ≤ 𝑘 < 𝑛 and 𝑘1 + 𝑘2 = 𝑘,
𝑒𝑘
2 +1) ln 𝑛 +
𝑡 𝑘1 , 𝑘2 , 𝑚, 𝑛) < 𝑒𝑘(𝑘
(𝑘 + 2𝑘2 + 1)
𝑘−𝑚+1
𝑘−𝑚+1 1

In Stage 1,
𝑛
𝑡((2,2), 3, )) < 6𝑒 ln 𝑛 𝑑 + 14𝑒 < 12 log 2 𝑛 𝑑 + 14𝑒.
𝑑
Theorem 6
This trivial two-stage algorithm identifies all positives in
12 log 2 𝑛 𝑑 + 14𝑒 + 3𝑑 group tests.
𝑛
𝑑
𝑛
𝑑
Furthermore, the decoding complexity is 𝑂( log + 𝑑).
40
2014/08/02
Two-stage for CGT
Concluding 3：
Theorem 4
The trivial two-stage algorithm which provided by De Bonis
et al. identifies all positives in
16 log 2 𝑛 𝑑 + 14𝑒 + 3𝑑 group tests.
𝑛
𝑑
𝑛
𝑑
Furthermore, its decoding complexity is 𝑂( log + 𝑑).
Theorem 6
By choosing a ((2,2), 3, 𝑛 𝑑 )-selector in the first stage, the
trivial two-stage algorithm identifies all positives in
12 log 2 𝑛 𝑑 + 14𝑒 + 3𝑑 group tests.
𝑛
𝑑
𝑛
𝑑
Furthermore, the decoding complexity is 𝑂( log + 𝑑).
41
2014/08/02
Reference
[1] Y. Bai, T. Huang, and K. Wang, Error-correcting pooling designs associated with some
distance-regular graphs, Discrete Appl. Math. 157 (2009) 1581-1585.
[2] C. J. Colbourn, Group testing for consecutive positives, Ann. Combin. 3 (1999) 37-41.
[3] A. De Bonis, L. Gasieniec, and U. Vaccaro, Optimal two-stage algorithms for group testing
problems, SIAM J. Comput. 34 (2005) 1253-1270.
[4] D.Z. Du and F. K. Hwang, Pooling Designs and Nonadaptive Group Testing - Important
Tools for DNA Sequencing, World Scientific (2006).
[5] A.G. D’yachkov, A.J. Macula, and P.A. Vilenkin, Nonadaptive and trivial two-stage group
testing with error-correcting 𝑑𝑒 -disjunt inclusion matrices, Entropy, Search, Complexity.
Bolyai Soc. Math. Stu. 16 (2007) 71-83.
[6] L. Lovasz, On the ratio of optimal integral and fractional covers, Discrete Math. 13 (1975)
383-390.
[7] B. Lv, K. Wang, and J. Guo, Error-tolerance pooling designs based on Johnson graphs,
Optim. Letters 8 (2014) 1161-1165.
[8] M. Muller and M. Jimbo, Consecutive positive detectable matrices and group testing for
consecutive positives, Discrete Math. 279 (2004) 369-381.
[9] S. K. Stein, Two combinatorial covering problems, J.Combin. Theory, Ser. A 16 (1974) 391397.
2014/08/02
Thank you for your attention.
2014/08/02
Pooling designs
Nonadaptive algorithm

A binary matrix 𝑀 is 𝒔𝒆 -disjunct if any 𝑠 + 1 columns
of 𝑀 with one designated, there are 𝑒 + 1 rows
intersecting the designated column and none of the other
𝑠 columns
𝐶
𝑠 columns
0
At least 𝑒 + 1 rows
⋯
⋯
1
1
1
0
0
0
1
1
0
0
⋮
⋮
⋮
⋯
⋯
⋯
0
0
0
⋯
⋯
0
0
⋯
6
2014/08/02
Two-stage for CGT
Construction of ( 𝒌𝟏 , 𝒌𝟐 , 𝒎, 𝒏)-selector：



ℋ 𝑋, ℱ ：a hypergraph
𝑋：a set of vertices
ℱ：a set of hyperedges 𝐸 (𝐸： a subset of 𝑋)
A cover 𝐶 of ℋ：𝐶 ⊆ 𝑋 and 𝐶 ∩ 𝐸 ≠ ∅, for all 𝐸 ∈ ℱ.
A cover of a properly defined hypergraph can produce a
( 𝑘1 , 𝑘2 , 𝑚, 𝑛)-selector
Lovasz-Stein Theorem (75’)
a greedy strategy provides a cover 𝐶 of ℋ with
𝑋
𝐶 ≤
1 + ln Δ
min 𝐸
𝐸∈𝐹
where Δ is the maximum vertex-degree
2014/08/02
Two-stage for CGT
Construction of ( 𝒌𝟏 , 𝒌𝟐 , 𝒎, 𝒏)-selector：

𝑋 ≔ {𝑥: binary vectors of length 𝑛 containing 𝑤 1’s.}

Let ℬ = {𝑒1 , 𝑒2 , … , 𝑒𝑘 } where 𝑒𝑖 is the row of 𝐼𝑘 satisfy only
𝑖-th entry equal to 1.

Suppose 𝛼 = (𝑎1 , 𝑎2 , … , 𝑎𝑘 ) ∈ ℬ
𝐸𝛼,(𝑆1 ,𝑆2 ) ≔ {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ 𝑋 ∶ 𝑥𝑗1 = 𝑎1 , 𝑥𝑗2 =
𝑎2 , … , 𝑥𝑗𝑘 = 𝑎𝑘 } if 𝑆1 ∪ 𝑆2 = {𝑗1 , 𝑗2 , … , 𝑗𝑘 } and 1 ≤ 𝑗1 < 𝑗2 <
⋯ < 𝑗𝑘 ≤ 𝑛.

An hyperedge 𝐸𝐴,(𝑆1 ,𝑆2 ) = 𝛼∈𝐴 𝐸𝛼,(𝑆1 ,𝑆2 ) , where 𝐴 ⊆ ℬ.
e.g. 𝑛 = 6, 𝑤 = 3 and 𝑘1 = 𝑘2 = 2
𝑋 = 1, 1, 1, 0, 0, 0 , 1, 1, 0, 1, 0, 0 ,· · · .
𝐸{𝑒1 ,𝑒3 },({3,4},{1,6}) = { 1,1,0,0,1,0 , (0,1,0,1,1,0)}
2014/08/02
Two-stage for CGT
Construction of ( 𝒌𝟏 , 𝒌𝟐 , 𝒎, 𝒏)-selector：

Define ℋ𝑟 𝑋, ℱ where ℱ ≔ {𝐸𝐴, 𝑆1 ,𝑆2 : 𝐴 ⊆ ℬ for 𝐴 = 𝑟,
𝑆1 ∩ 𝑆2 = ∅, and 𝑆1 consists of 𝑘1 consecutive numbers and
𝑆2 = 𝑘2 }.

A cover 𝐶 of ℋ𝑘−𝑚+1 (𝑋, ℱ)：


All elements in 𝐶 of ℋ𝑘−𝑚+1 ⟹ rows of a binary matrix 𝑀.
Example：
𝐶 = 1,1,1,0,0,0 , 1,0,0,1,1,0 , 1,0,1,0,0,1 , …
𝑀=
1
1
1
1
0
0
1
0
1
0
1
0
0
1
0
0
0
1
⋮
⋮

Such a matrix 𝑀 is ( 𝑘1 , 𝑘2 , 𝑚, 𝑛)-selector.