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Error-correcting Pooling Designs and Group T esting for Consecutive Positives. Advisor : Huilan Chang Student : Yi- Tsz Tsai. Department of Applied Mathematics National Kaohsiung University. Outline. Error-correcting pooling designs Constructed from vectors - PowerPoint PPT Presentation
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Error-correcting Pooling Designsand
Group Testing for Consecutive Positives
Advisor : Huilan Chang
Student : Yi-Tsz Tsai
2014/08/02
Department of Applied MathematicsNational Kaohsiung University
Outline Error-correcting pooling designs
Constructed from vectors
Constructed from distance-regular graph
Two-stage algorithm for Group testing of consecutive positives
2014/08/02
1
Classical group testing items where each item is either positive or negative. Information : at most positives. () Goal : identify all positives by group tests.
Positive
Negative
Pooling designs2014/08/02
2
2014/08/02
ypes 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.
Pooling designs
3
Nonadaptive algorithm Binary matrix representation :
Rows are tests. Columns are items. An entry if test contains item
2014/08/02
Pooling designs
1 0 0 1 0 0 10 1 0 1 1 1 00 0 1 1 1 0 11 0 1 0 0 1 1
Testing outcome
𝑡1𝑡 2𝑡 3𝑡 4
4
Nonadaptive algorithm A binary matrix is -disjunct if any columns of with
one designated, there is a row intersecting the designated column and none of the other columns.
1 0 ⋯ 0
columns
At least row
𝐶0
⋯⋯ ⋯
⋮
2014/08/02
Pooling designs
5
Nonadaptive algorithm A binary matrix is -disjunct if any columns of with
one designated, there are rows intersecting the designated column and none of the other columns
2014/08/02
columns
At least rows
𝐶0
⋯⋯ ⋯
Pooling designs
6
Nonadaptive algorithm Note :
An -disjunct matrix is also called -disjunct. An -disjunct matrix is fully -disjunct
if it is not -disjunct whenever or . Application :
A -disjunct matrix is -error-correcting.
2014/08/02
Pooling designs
7
Error-correcting pooling designs
Constructed from vectors Constructed from distance-regular graph
Construction2014/08/02
8
Error-correcting pooling designs
Constructed from vectors A -ary vector is a vector whose entries are from . The weight of a vector of its nonzero entries. all -ary vectors of length and weight .
2014/08/02
Construction - sequences
9
2014/08/02
Definition (D’ychakov et al., 05’)
Let and .: binary matrix by rows indexed and columns indexed by .For and , iff .
: the -th entry of . : if for , whenever . Example : In ,
Construction - sequences
(1 , 2 ,0,0 ,1 , 0)≺(1 ,2 ,2,0 ,1 ,0)(1 , 2 ,2,0 ,1 , 0)
(1 ,2 ,0,0 ,1 ,0)10
1
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Theorem (D’ychakov et al., 05’)
Let and . is fully -disjunct where .
Construction - sequences
How about “ for some ” ?
11
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Definition 1
Let and .: binary matrix by rows indexed and columns indexed by .For and , iff .
if and for otherwise. Example :
Construction - sequences
12
2014/08/02
Definition 1
Let and .: binary matrix by rows indexed and columns indexed by .For and , iff .
if and for otherwise. Example :
Construction - sequences
13
Example : iff
𝜋 2(1 ;2,3 , 4)
2014/08/02
Construction - sequences
0 01 10 0⋯⋯⋯ 0 0
⋯⋯⋯
(1 1 0 0)(1 0 1 0)(1 0 0 1)(0 1 1 0)
(0 0 2 2)(0 20 2)
⋯⋯⋯
0 10 111⋯⋯⋯ 0 01 00 111⋯⋯⋯ 0 00 11 011⋯⋯⋯ 0 0
0 0 00 01⋯⋯⋯ 1 1
⋯⋯⋯
⋯⋯⋯
0 0 00 11⋯⋯⋯ 0 0
(1 11 0)
(1 1 0 1)
(1 0 1 1)
(0 1 1 1)
(1 12 0)
(1 1 0 2)(2 0 2 2)
(2 0 2 2)
14
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Theorem 1
Let and .Then is -disjunct, where
.
Construction - sequences
Theorem 2
Let and .Then is -disjunct, where
.
Our result:
15
Goal:How many that and ?
designated-ary vector of weight
-ary vector of weight 𝛽0𝛽1
𝛼1
𝛽𝑠⋯⋯
𝛼2
𝛼(𝑛𝑑)𝑞𝑑
⋯⋯⋯
⋯⋯ 𝛽(𝑛𝑘)𝑞𝑘⋯⋯
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Analysis:Construction - sequences
16
Step1 : Find the lower bound of the number of -ary vectors of weight satisfying and for each .
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Analysis: (Theorem 2)Construction - sequences
( … … … … )
( … … … … )
( … … … … )
( … … … … )
( … … … … )
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
≠≠
≠
⋯
{ : and either or and }
17
𝑟 −|𝑋 0|
Step1 : Find the lower bound of the number of -ary vectors of weight satisfying and for each .
2014/08/02
Analysis: (Theorem 2)Construction - sequences
( … … … … )
( … … … … )
( … … … … )
( … … … … )
( … … … … )
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
≠≠
≠
⋯
¿
{ : and either or and }
17
≠ 0
Step1 : Find the lower bound of the number of -ary vectors of weight satisfying and for each .
(𝑘−|𝑋 0|𝑟−|𝑋 0|)≥(𝑘−𝑠
𝑟 −𝑠)
2014/08/02
Analysis: (Theorem 2)Construction - sequences
( … … 0 0 … 0 0 … 0 )
( … … … … )
( … … … … )
( … … … … )
( … … … … )
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
≠≠
≠
{ : and either or and }
𝑟 −|𝑋 0|
¿
⋯
17
≠ 0
2014/08/02
( … … … … )
Construction - sequences
Step2 : Find
• Fixed , choose satisfying wherever
Analysis: (Theorem 2)
( … … 0 0 … 0 0 … 0 )
( … … … … )
( … … … … )
( … … … … )
( … … … … )
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
≠≠
≠
¿
⋯
18
≠ 0
2014/08/02
( … … … … )
Construction - sequences
Step2 : Find
• Fixed , choose satisfying wherever
Analysis: (Theorem 2)
( … … 0 0 … 0 0 … 0 )
( … … … … )
( … … … … )
( … … … … )
( … … … … )
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
≠≠
≠
¿
⋯
(𝑑−𝑟 )
¿¿
≠
18
≠ 0
2014/08/02
( … … … 0 … 0 )
Construction - sequences
Step2 : Find
• Fixed , choose satisfying wherever
Analysis: (Theorem 2)
( … … 0 0 … 0 0 … 0 )
( … … … … )
( … … … … )
( … … … … )
( … … … … )
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿
≠≠
≠
¿
⋯
(𝑑−𝑟 )(𝑛−𝑟𝑑−𝑟 )(𝑞−𝑠−1)𝑑− 𝑟
≠
¿¿
18
≠ 0
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Construction - sequences
Concluding 1:
(1)
(2)
(3) Our results :by “intersecting relation”
For given and , if , then , where (2) or (3).
D’ychakov et al. (2005) :by “containment relation”
19
Error-correcting pooling designs
Constructed from distance-regular graph The Johnson graph is defined on such that two
vertices and are adjacent iff . Binary matrix with columns and rows indexed by
-cliques.
2014/08/02
Construction – Johnson graph
20
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-clique : For a connected graph :
an -subset of is a -clique if any two vertices in are at distance .
In Johnson graph :a -clique with size is a collection of disjoint -subsets of .
Example :• if
Construction – Johnson graph
123124
125
234235
456
245
236
⋯
⋯21
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-clique : For a connected graph :
an -subset of is a -clique if any two vertices in are at distance .
In Johnson graph :a -clique with size is a collection of disjoint -subsets of .
Example : 123124
125
234235
456
245
236
• if
• is a -clique of size .
Construction – Johnson graph
⋯
⋯21
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Definition (Bai et al., 09’)
Let and .: binary matrix with rows indexed by -clique with size and columns indexed by -clique with size such that iff .
Construction – Johnson graph
( {1,2 } )
⋯⋯⋯
1 110⋯⋯⋯ 0⋯ ⋯
( {1,2 }{3,5 }{4,6 })(
{1,2 }{3,6 }{4,5 }) ( {1,6 }
{2,5 }{3,4 })( {1,2 }
{3,4 }{5,6 })
( {1,3 } )
( {5,6 } )
⋯
0 0 01⋯⋯⋯ 0
0 0 01⋯⋯⋯ 0
( {1,3 }{2,4 }{5,6 })𝑀 (6 ,2 , 1 ,3)
( {4,6 } ) 0 10 0⋯⋯⋯ 022
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Theorem (Bai et al., 09’)
Let and . is fully -disjunct where .
How about “ iff ” ?
Construction – Johnson graph
23
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Definition 2
Let and .: binary matrix with rows indexed by -clique with size and columns indexed by -clique with size such that iff .
Construction – Johnson graph
𝑀 (1 ;6 , 2 ,2 ,3)
( {1,2 }{3,4 })
⋯⋯⋯
0 11 0⋯⋯⋯ 1⋯ ⋯
( {1,2 }{3,5 }{4,6 })(
{1,2 }{3,6 }{4,5 }) ( {1,6 }
{2,5 }{3,4 })( {1,2 }
{3,4 }{5,6 })
( {1,2 }{3,5 })
( {3,6 }{4,5 })
⋯
1 01 0⋯⋯⋯ 0
0 01 0⋯⋯⋯ 0
( {1,3 }{2,4 }{5,6 })
24
2014/08/02
Construction – Johnson graph
Theorem (Lv et al., 14’)
Let and
.
Then is -disjunct, where
.
25
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Theorem 3
Let and .
Then is -disjunct, where .
Our result:Construction – Johnson graph
26
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Construction – Johnson graph
Table 1 : Some comparisons of error-tolerance capabilities of
Lv et al. (14’) : Our result :
Concluding 2:
27
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Our results for error-correcting capabilities:
Construction by intersecting relation : -ary vectors
-cliques of Johnson graph
Conclusion-Pooling designs
28
Two-stage algorithm for group testing of consecutive positives
Two-stage for CGT2014/08/02
29
2014/08/02
Group testing of consecutive positives A set of objects satisfying the linear order .
Positives are consecutive under .Information : at most positives. ()
Motivation: applications in DNA sequencing.
Two-stage for CGT
30
2014/08/02
Group testing of consecutive positives Nonadaptive algorithm :
Begin by partition into partsEach part contains consecutive items.
All positive items are contained in at most two parts.
Two-stage for CGT
31
2014/08/02
Group testing of consecutive positives Nonadaptive algorithm :
Begin by partition into parts.Each part contains consecutive items.
All positive items are contained in at most two parts. Colburn (99’) :
Gray code tests. Mller and Jimbo (04’) :
consecutive positive detectable matrices
tests.
Two-stage for CGT
32
Multi-stage algorithm Multi-stage algorithm : Stages are sequential and
all tests in a stage are nonadaptive. Example : and at most positives.
Especially called a “trivial two-stage algorithm”
if Stage 2 = identity matrix.
2014/08/02
Two-stage for CGT
33
2014/08/02
Definition (De Bonis et al., 05’)
Given , a binary matrix is a -selector if any submatrix of
obtained by choosing out of arbitrary columns of
contains at least distinct rows of the identity matrix .
-selector :Two-stage for CGT
At least rows of
Arbitrary columns
⋯ ⋯10000100
01010100
-selector
001034
2014/08/02
Theorem (De Bonis et al., 05’)
Let , there exists a -selector of size , with
Trivial two-stage algorithm (De Bonis et al., 05’):Two-stage for CGT
Partition into parts .
-selector
Stage 1
𝑋 1𝑋 2𝑋 3 𝑋 ⌈ 𝑛/𝑑⌉…
Identity
Stage 2
𝑋 𝑖𝑋 𝑗𝑋𝑘
At most 3 parts left
35
2014/08/02
Trivial two-stage algorithm (De Bonis et al., 05’) :Two-stage for CGT
Theorem 4
This trivial two-stage algorithm identifies all positives in
group tests.
Furthermore, its decoding complexity is .
-selectors were not specifically introduced to deal with the group testing of consecutive positives.
Next, we consider its variation
-selectors
36
2014/08/02
Definition 3
For and ,
A binary matrix is a -selector if any submatrix of
obtained by choosing consecutive columns and other
arbitrary columns contains at least distinct rows of the
identity matrix .
Two-stage for CGT
11000010
01010100
arbitraryconsecutive -selector
rows of (in the submatrix)
⋯ ⋯
-selector :
37
2014/08/02
Partition into parts . Stage 1 :
Use a -selector as a pooling design where the -th column associated with .
Discard each part contained in any negative test. Stage 2 : Identity matrix.
Our two-stage algorithm:Two-stage for CGT
-selector
Stage 1
𝑋 1𝑋 2𝑋 3 𝑋 ⌈ 𝑛/𝑑⌉…
Identity
Stage 2
𝑋 𝑖𝑋 𝑗𝑋𝑘
At most ? parts left
38
2014/08/02
Lemma
After using a -selector, there remain at most 3 parts in
Stage 1 and their union contains all positive items.
Two-stage for CGT
Note : the min. number of rows among all -selectors.
Stage 2 : Test each item in the remaining parts individually
There are tests.
Next, the upper bound of ?
39
2014/08/02
Theorem 5 ( by Lovsz-Stein Theorem)
Let and ,
Two-stage for CGT
In Stage 1, .Theorem 6
This trivial two-stage algorithm identifies all positives in
group tests.
Furthermore, the decoding complexity is . 40
2014/08/02
Concluding 3:Theorem 4
The trivial two-stage algorithm which provided by De Bonis et al. identifies all positives in
group tests.
Furthermore, its decoding complexity is .
Theorem 6
By choosing a -selector in the first stage, the trivial two-stage algorithm identifies all positives in
group tests.
Furthermore, the decoding complexity is .
Two-stage for CGT
41
[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 -disjunt inclusion matrices, Entropy, Search, Complexity. Bolyai Soc. Math. Stu. 16 (2007) 71-83.
[6] L. Lovsz, 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. Mller 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) 391-397.
2014/08/02
Reference
Thank you for your attention.
2014/08/02
Nonadaptive algorithm A binary matrix is -disjunct if any columns of with
one designated, there are rows intersecting the designated column and none of the other columns
2014/08/02
columns
At least rows
𝐶0
⋯⋯ ⋯
Pooling designs
6
2014/08/02
Two-stage for CGT
Construction of -selector: : a hypergraph
: a set of vertices : a set of hyperedges ( : a subset of )
A cover of : and , for all .
A cover of a properly defined hypergraph can produce a -selector
Lovsz-Stein Theorem (75’)
a greedy strategy provides a cover of with
where is the maximum vertex-degree
2014/08/02
Two-stage for CGT
Construction of -selector: binary vectors of length containing ’s.
Let where is the row of satisfy only -th entry equal to .
Suppose
if and .
An hyperedge , where .
e.g. and
2014/08/02
Two-stage for CGT
Construction of -selector: Define where for , , and consists of consecutive numbers
and .
A cover of : All elements in of rows of a binary matrix .
Example :
Such a matrix is -selector.
𝑀=¿
1 1 1 0 0 0 1 0 0 1 1 01 0 1 0 0 1