Error-correcting Pooling Designs and Group T esting for Consecutive Positives

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


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


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 ,


(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

2014/08/02

Theorem (D’ychakov et al., 05’)

Let and . is fully -disjunct where .


How about “ for some ” ？

11

2014/08/02

Definition 1


if and for otherwise. Example ：


12

2014/08/02

Definition 1


if and for otherwise. Example ：


13

Example ： iff

𝜋 2(1 ;2,3 , 4)

2014/08/02


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

2014/08/02

Theorem 1

Let and .Then is -disjunct, where

.


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⋯⋯

2014/08/02

Analysis：Construction - sequences

16

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

𝑟 −|𝑋 0|


2014/08/02


( … … … … )

( … … … … )

( … … … … )

( … … … … )

( … … … … )

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

≠≠

≠

⋯

¿


17

≠ 0


(𝑘−|𝑋 0|𝑟−|𝑋 0|)≥(𝑘−𝑠

𝑟 −𝑠)

2014/08/02


( … … 0 0 … 0 0 … 0 )

( … … … … )

( … … … … )

( … … … … )

( … … … … )

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

≠≠

≠


𝑟 −|𝑋 0|

¿

⋯

17

≠ 0

2014/08/02

( … … … … )


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

( … … … … )


Step2 ： Find



( … … 0 0 … 0 0 … 0 )

( … … … … )

( … … … … )

( … … … … )

( … … … … )

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

≠≠

≠

¿

⋯

(𝑑−𝑟 )

¿¿

≠

18

≠ 0

2014/08/02

( … … … 0 … 0 )


Step2 ： Find



( … … 0 0 … 0 0 … 0 )

( … … … … )

( … … … … )

( … … … … )

( … … … … )

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

𝑎1𝑎2𝑎3 𝑎¿ 𝑋0∨¿¿

≠≠

≠

¿

⋯

(𝑑−𝑟 )(𝑛−𝑟𝑑−𝑟 )(𝑞−𝑠−1)𝑑− 𝑟

≠

¿¿

18

≠ 0

2014/08/02


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


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

2014/08/02

-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


123124

125

234235

456

245

236

⋯

⋯21

2014/08/02

-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 .


⋯

⋯21

2014/08/02

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 .


( {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

2014/08/02

Theorem (Bai et al., 09’)

Let and . is fully -disjunct where .

How about “ iff ” ？


23

2014/08/02

Definition 2

Let and .： binary matrix with rows indexed by -clique with size and columns indexed by -clique with size such that iff .


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


Theorem (Lv et al., 14’)

Let and

.

Then is -disjunct, where

.

25

2014/08/02

Theorem 3

Let and .

Then is -disjunct, where .

Our result：Construction – Johnson graph

26

2014/08/02


Table 1 ： Some comparisons of error-tolerance capabilities of

Lv et al. (14’) ： Our result ：

Concluding 2：

27

2014/08/02

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


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

Documents

Error-correcting Pooling Designs and Group T esting for Consecutive Positives