Transforming an error-tolerant separable matrix to an error-tolerant disjunct matrix

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Discrete Applied Mathematics 157 (2009) 387–390

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Note

Transforming an error-tolerant separable matrix to an error-tolerant

disjunct matrix

I

Hong-Bin Chen

a

, Yongxi Cheng

b

, Qian He

c,∗

, Chongchong Zhong

c

a_{Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30050, Taiwan} b_{Institute for Theoretical Computer Science, Tsinghua University, Beijing, 100084, China} c_{Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China}

a r t i c l e i n f o

Article history:

Received 7 September 2007 Received in revised form 29 May 2008 Accepted 3 June 2008

Available online 9 July 2008

Keywords:

Error-tolerant Separable matrices Disjunct matrices

a b s t r a c t

Recently, Chen and Hwang [H.B. Chen, F.K. Hwang, Exploring the missing link among d-separable, d-separable and d-disjunct matrices, Discrete Applied Mathematics 133 (2007) 662–664] provided a method for transforming a separable matrix to a disjunct matrix. In [D.Z. Du, F.K. Hwang, Pooling Designs and Nonadaptive Group Testing — Important Tools for DNA Sequencing, World Scientific, 2006], Du and Hwang attempted to extend this result to its error-tolerant version; unfortunately, they gave an incorrect extension. This note gives a solution to this problem.

1. Introduction

Let M be a

(

0

,

1

)

matrix. For any set S of columns of M, U

(

S

)

will denote the union of the row indices of 1-entries of all columns in S. When S is the singleton set

{

C

}

, we abuse the notation by writing U

(

S

)

simply as C . M is called d-separable if for any two distinct d-sets S and S0_{of columns, U}

₍

_S

_{) 6=}

_U

₍

_S0

₎

_{. M is called d-separable if the restrictions}

_|

_S

_{| =}

_{d and}

_|

_S0

_{| =}

_d above are changed to

|

S

| ≤

d and

|

S0

_{| ≤}

_{d, respectively. Finally, M is called d-disjunct if for any d-set S of columns and} any column C not in S, C is not contained in U

(

S

)

. These three properties of

(

0

,

1

)

matrices have been widely studied in the literature of nonadaptive group testing designs (pooling designs), which have applications in DNA screening [2–7].

It has long been known that d-disjunctness implies d-separability which in turn implies d-separability [3, Chapter 2]. Recently, Chen and Hwang [1] found a way to construct a disjunct matrix from a separable matrix to complete the cycle of implications.

Theorem 1.1 (Chen and Hwang [1]). Suppose M is a 2d-separable matrix. Then one can construct a d-disjunct matrix by adding at most one row to M.

The notions of d-separability, d-separability and d-disjunctness have error-tolerant versions. A

(

0

,

1

)

matrix M is called

(

d

;

z

)

-separable if

|

U

(

S

)4

U

(

S0

)| ≥

z for any two d-sets of columns of M. It is

(

d

;

z

)

-separable if the restriction of d-sets is changed to two sets each with at most d elements. Finally, M is

(

d

;

z

)

-disjunct if for any d-set S of columns and any column

I _{He and Zhong are supported by Science Technology Commission of Shanghai Municipality (Grant 06ZR14048) and Chinese Ministry of Education (Grant}

108056).

∗_{Corresponding author. Fax: +86 21 54743152.}

E-mail addresses:[email protected](H.-B. Chen),[email protected](Y. Cheng),[email protected],[email protected](Q. He), [email protected](C. Zhong).

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388 H.-B. Chen et al. / Discrete Applied Mathematics 157 (2009) 387–390

C not in S,

|

C

\

U

(

S

)| ≥

z. Note that the variable z represents some redundancy for tolerating errors [3]. For z

=

1, the error-tolerant version is reduced to the original version.

Du and Hwang attempted to extendTheorem 1.1to its error-tolerant version.

Theorem 1.2 ([3, Theorem 2.7.6]). Suppose M is a

(

2d

;

z

)

-separable matrix. Then one can obtain a

(

d

;

z

)

-disjunct matrix by adding at most z rows to M.

ByTheorem 1.2, Du and Hwang obtained the following corollary.

Corollary 1.3 ([3, Theorem 2.7.7]). A

(

d

;

2z

)

-separable matrix can be obtained from a

(

2d

;

z

)

-separable matrix by adding at most z rows.

Unfortunately,Theorem 1.2is incorrect; thusCorollary 1.3is incorrect as seen from the following counter-example. Let

M1

=







1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1





 .

It is easily verified that M1is

(

2

;

2

)

-separable. We now show that adding two rows to M1cannot produce a

(

1

;

2

)

-disjunct matrix.

Let C1

,

C2

,

C3

,

C4denote the four columns of M1. Suppose we set C

=

Ciand S

= {

Cj

}

, i

6=

j. Then we need two rows each containing Cibut not Cj. One such row is already provided by M1. So we need one

(

1

,

0

)

-pair in a new row. Since this is required for each pair of

(

i

,

j

)

with i

6=

j, there are 4

×

3

=

12 choices of

(

i

,

j

)

pairs and each such pair needs a

(

1

,

0

)

-pair in a new row; or equivalently, we need the new rows to provide twelve such

(

1

,

0

)

-pairs. But one new row can provide at most four

(

1

,

0

)

-pairs (achieved by a row with two 1-entries and two 0-entries). So two new rows are not sufficient for providing the twelve

(

1

,

0

)

-pairs required by the

(

1

;

2

)

-disjunctness property.

In this note we give a correct version ofTheorem 1.2, and obtain a more rigorous statement ofTheorem 1.1.

2. Main results

Lemma 2.1 ([3, Lemma 2.1.1]). Suppose M is a d-separable matrix with n columns where d

<

n; then it is k-separable for every positive integer k

≤

d.

Note that the condition d

<

n inLemma 2.1is necessary as seen from the following example: Let

M2

=

1 1 0 1 0 1 0 1 1

!

.

M2is trivially 3-separable. But it is not 2-separable, as the union of any pair of its columns is identical. We now generalizeLemma 2.1to an error-tolerant version.

Lemma 2.2. If a matrix M with n columns is

(

d

;

z

)

-separable for d

<

n, then it is

(

k

;

z

)

-separable for every positive integer k

≤

d.

Proof. It suffices to prove that M is

(

d

−

1

;

z

)

-separable. Assume that M is not

(

d

−

1

;

z

)

-separable. Then there exist two distinct sets S and S0each consisting of d

−

1 columns of M such that

|

U

(

S

)4

U

(

S0

)| <

z.

If

|

S

\

S0

_{| = |}

_S0

_\

_S

_{| ≥}

_{2, then there must exist a pair of columns}

₍

_C

x

,

Cy

)

such that Cx

∈

S

\

S0and Cy

∈

S0

\

S. It is easy to see that

|

U

(

S

∪ {

Cy

}

)4

U

(

S0

∪ {

Cx

}

)| ≤ |

U

(

S

∪ {

Cy

}

)4

U

(

S0

)| ≤ |

U

(

S

)4

U

(

S0

)|.

This violates the

(

d

;

z

)

-separability of M, as desired.

Now consider the case of

|

S

\

S0

| = |

S0

\

S

| =

1. It is obvious that

|

S

∪

S0

| =

d. Thanks to d

<

n, we can take a column C

of M which is in neither S nor S0_{. It is easily seen that}

_|

_U

₍

_S

_{∪ {}

_C

_}

₎₄

_U

₍

_S0

_{∪ {}

_C

_}

_{)| ≤ |}

_U

₍

_S

₎₄

_U

₍

_S0

_{)| <}

_{z. This contradicts the}

(

d

;

z

)

-separability of M, completing the proof.

We are ready to give a correct version ofTheorem 1.2.

Theorem 2.3. Suppose M is a

(

2d

;

z

)

-separable matrix with n columns where n

≥

2d

+

1. Then one can obtain a

(

d

; d

z

/

2

e

)

-disjunct matrix by adding at most

d

z

/

2

e

rows to M.

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H.-B. Chen et al. / Discrete Applied Mathematics 157 (2009) 387–390 389 Proof. Suppose M is not

(

d

; d

z

/

2

e

)

-disjunct. Then there exist a column C and a set S of d other columns such that

|

C

\

U

(

S

)| < d

z

/

2

e

. By adding at most

d

z

/

2

e

rows to M such that each row has a 1-entry at column C and 0-entries at all columns in S, we can obtain

|

C

\

U

(

S

)| ≥ d

z

/

2

e

. Of course, there may exist another pair

(

C0

_,

_S0

₎

_{where C}0_{is a column and}

S0is a set of d columns other than C0, such that

|

C0

\

U

(

S0

)| < d

z

/

2

e

in M. Then we break it up by using those

d

z

/

2

e

rows in the same fashion. What we need to show is that this procedure is not self-conflicting, i.e., there do not exist two pairs

(

C

,

S

)

and

(

C0

_,

_S0

₎

_{such that}

_|

_C

_\

_U

₍

_S

_{)| < d}

_z

_/

₂

_e

_{, yet on the other hand C}

_∈

_S0_while

_|

_C0

_\

_U

₍

_S0

_{)| < d}

_z

_/

₂

_e

_.

Suppose to the contrary that there exist two pairs

(

C

,

S

)

and

(

C0

_,

_S0

₎

_{in M as described above with}

_|

_S

_{| = |}

_S0

_{| =}

_{d. Define}

S0

= {

C0

} ∪

S

∪

S0, S1

=

S0

\ {

C

}

, and S2

=

S0

\ {

C0

}

. Let s

= |

S0

|

; then s

≤

2d

+

1 and

|

S1

| = |

S2

| =

s

−

1

≤

2d.

Note that S1

6=

S2, but they have the same cardinality which is less than 2d

+

1. We now show the symmetric difference of U

(

S1

)

and U

(

S2

)

is less than z, thus violating the assumption of

(

2d

;

z

)

-separability.

Since the only column in S1but not in S2is C0and

|

C0

\

U

(

S0

)| < d

z

/

2

e

, we have

|

U

(

S1

) \

U

(

S2

)| < d

z

/

2

e

.

(1)

Similarly, we can obtain

|

U

(

S2

) \

U

(

S1

)| < d

z

/

2

e

.

(2)

Eq.(1)along with Eq.(2)gives

|

U

(

S1

)4

U

(

S2

)| <

z, implying that M is not

(

s

−

1

;

z

)

-separable. This contradictsLemma 2.2 and so we have completed the proof.

Corollary 2.4. Suppose M is a 2d-separable matrix with n columns where n

≥

2d

+

1. Then one can obtain a d-disjunct matrix

by adding at most one row to M.

Proof. It follows fromTheorem 2.3on setting z

=

1

.

Corollary 2.4is a more rigorous version ofTheorem 1.1. The following example shows the necessity of the extra condition

n

≥

2d

+

1 inCorollary 2.4. Let M3

=







1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1





 .

Then M3is trivially 4-separable; but it can be easily verified that no row can be added to M3to make it 2-disjunct. Similarly, any matrix with 2d columns is trivially

(

2d

;

z

)

-separable and one does not expect that adding

d

z

/

2

e

rows to an arbitrary matrix with 2d columns would make it

(

d

; d

z

/

2

e

)

-disjunct. To see a specific counter-example, note that M1is trivially a

(

4

;

4

)

-separable matrix; but adding two rows does not make it a

(

2

;

2

)

-disjunct matrix – it is even not

(

1

;

2

)

-disjunct as indicated at the end of Section1.

Corollary 2.5. Suppose M is a

(

2d

;

z

)

≥

2d

+

1. Then, for any positive integer

k

≤ d

z

/

2

e

, one can obtain a

(

d

;

k

)

-disjunct matrix by adding at most k rows to M.

Proof. The proof ofTheorem 2.3shows that there do not exist two pairs

(

C

,

S

)

and

(

C0

_,

_S0

₎

_{such that}

_|

_C

_\

_U

₍

_S

_{)| < d}

_z

_/

₂

_e

_, yet on the other hand C

∈

S0while

|

C0

\

U

(

S0

)| < d

z

/

2

e

. In fact, the term

d

z

/

2

e

can be replaced by any positive integer

k which satisfies the symmetric difference of U

(

S1

)

and U

(

S2

)

is less than z. Therefore, for any k

≤ d

z

/

2

e

, we can obtain a

(

d

;

k

)

-disjunct matrix by adding at most k rows to M in the same fashion.

The following equivalence relation is given in [3] without giving a proof. We now give a proof and use the equivalence relation to obtain a stronger result.

Lemma 2.6 ([3, Lemma 2.7.5]). A matrix M is

(

d

;

z

)

-separable if and only if it is

(

d

;

z

)

-separable and

(

d

−

1

;

z

)

-disjunct. Proof. Suppose M is

(

d

;

z

)

-separable but not

(

d

−

1

;

z

)

-disjunct, in other words, there exists a set S of d

−

1 columns other than a column C such that

|

C

\

U

(

S

)| ≤

z. Then it is easy to see that

|

U

(

S

∪ {

C

}

)4

U

(

S

)| = |

U

(

S

∪ {

C

}

) \

U

(

S

)| ≤

z, a

contradiction to

(

d

;

z

)

-separability. Thus, M is

(

d

−

1

;

z

)

-disjunct and

(

d

;

z

)

-separable trivially.

Let M be

(

d

;

z

)

-separable and

(

d

−

1

;

z

)

-disjunct. It suffices to show that

|

U

(

X

)4

U

(

Y

)| ≥

z for any two sets X , Y of at

most d columns. If

|

X

| = |

Y

| ≤

d, then

|

U

(

X

)4

U

(

Y

)| ≥

z by

(

d

;

z

)

-separability andLemma 2.2. Assume

|

X

|

< |

Y

| ≤

d; then

there exists a column Cy

∈

Y but not in X . By

(

d

−

1

;

z

)

-disjunctness, we obtain

|

Cy

\

U

(

X

)| ≥

z; hence

|

U

(

X

)4

U

(

Y

)| ≥

z. This completes the proof.

ByLemmas 2.6and2.2, we extendCorollary 2.5to a stronger version.

Corollary 2.7. Suppose M is a

(

2d

;

z

)

≥

2d

+

1. Then, for any positive integer

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390 H.-B. Chen et al. / Discrete Applied Mathematics 157 (2009) 387–390

3. Concluding remarks

The following remarks demonstrate the optimality of our results.

Remark 1. The constraint k

≤ d

z

/

2

e

inCorollary 2.5is necessary if we want the number of rows added to be independent of n and d. To see a specific example, consider that M is an

(

n

d

z

/

2

e

) ×

n matrix such that each column has

d

z

/

2

e

1-entries and any two columns have no intersection. Then, M is

(

2d

;

z

)

-separable. Since every column has only

d

z

/

2

e

1-entries, to make M

(

d

;

k

)

-disjunct by adding rows, the rows added must form a

(

d

;

k

− d

z

/

2

e

)

-disjunct submatrix when k

> d

z

/

2

e

. In this case, the minimum number of rows required would depend on n

,

d and k

− d

z

/

2

e

.

Remark 2. Let N be a

(

0

,

1

)

matrix of constant row sum 1 and constant column sum z and let M be obtained from N by adding one zero column. It is easy to verify that M is

(

2d

;

z

)

-separable. Since there is a zero column in M, we cannot obtain from M a

(

d

;

k

)

-disjunct matrix by adding less than k rows. This shows that the bound on the number of additional rows given inCorollary 2.5is optimal in this sense.

References

[1] H.B. Chen, F.K. Hwang, Exploring the missing link among d-separable, d-separable and d-disjunct matrices, Discrete Applied Mathematics 133 (2007) 662–664.

[2] D.Z. Du, F.K. Hwang, Combinatorical Group Testing and its Applications, 2nd edition, World Scientific, 1999.

[3] D.Z. Du, F.K. Hwang, Pooling Designs and Nonadaptive Group Testing — Important Tools for DNA Sequencing, World Scientific, 2006.

[4] P. Erdős, P. Frankl, Z. Füredi, Families of finite sets in which no set is covered by the union of r others, Israel Journal of Mathematics 51 (1985) 79–89. [5] F.K. Hwang, V.T. Sós, Non-adaptive hypergeometric group testing, Studia Scientiarum Mathematicarum Hungarica 22 (1987) 257–263.

[6] A.J. Macula, A simple construction of d-disjunct matrices with certain constant weights, Discrete Mathematics 162 (1996) 311–312. [7] A.J. Macula, Error-correcting nonadaptive group testing with de_{-disjunct matrices, Discrete Applied Mathematics 80 (1997) 217–222.}