Volume 13, Number 4, 2006 © Mary Ann Liebert, Inc. Pp. 990–995
New Construction for Transversal Design
DING-ZHU DU,1 F.K. HWANG,2 WEILI WU,1 and TAIEB ZNATI3
ABSTRACT
The study of gene functions requires the development of a DNA library of high quality
through much of testing and screening. Pooling design is a mathematical tool to reduce the
number of tests for DNA library screening. The transversal design is a special type of pooling
design, which is good in implementation. In this paper, we present a new construction for
transversal designs. We will also extend our construction to the error-tolerant case.
Key words: pooling design, transversal design, new construction.
1. INTRODUCTION
A
recent important development in biology is the success of Human Genome Project. This project was done with a great deal of help from computer technology, which made computational biology a hot interdisciplinary research area between molecular biology, computer science, and mathematics. As the technology for obtaining sequenced genome data matures, more and more sequenced genome data are available to the scientific research community, so that the study of gene functions has become a popular research direction. Such a study is supported by a high quality DNA library which is usually obtained through much testing and screening. Therefore, the efficiency of testing and screening becomes very important. Pooling design is a mathematical tool to reduce the number of tests in DNA library screening (D’yachkov et al., 2001; Farach et al., 1997). For example the Life Science Division of Los Alamos National Laboratories in 1998 (Marathe et al., 2000) was dealing with 220,000 clones. Testing those clones individually requires 220,000 tests. However, they used only 376 tests with pooling designs. Pooling design is also called nonadaptive group testing. Given a set of n items with at most d positive ones, group testing tests subsets of items, called pools, instead of individual items. For example, in the above mentioned testing at Los Alamos National Laboratories, each pool contains about 5,000 clones. The outcome of a test on a pool is positive if the pool contains a positive item and is negative otherwise.The technology of group testing was initiated from a Wasserman-type blood test in World War II. Since then, many constructions have been developed in the literature (Du and Hwang, 1999, unpublished). A group testing algorithm is said to be nonadaptive if all tests are arranged in a single round, that is, if no information on test outcomes is available for determining the composition of another test. A pooling design is said to be transversal if it can be divided into disjoint families, each of which is a partition of all items such that pools in different parts are disjoint.
1Department of Computer Science, University of Texas at Dallas, Richardson, TX 75083. 2Department of Applied Mathematics, National Chiaotung University, Hsing Chu, Taiwan, ROC. 3Department of Computer Science, University of Pittsbugh, Pittsburgh, PA 15215.
In this paper, we present a new construction for transversal design. To identify n items with at most d positive ones, our construction gives a transversal design with at most (2+ o(1))log(d log n)dlog n )2tests, which is superior to all previously known transversal designs. This construction can also be easily extended to the error-tolerant case, which is an important topic in pooling designs (Ngo and Du, 2000, 2002; Hwang, 2003; Macula, 1997; Wu et al., 2003, submitted).
Transversal designs are used very frequently in practice because implementation is easy and their per-formance is quite good. Therefore, our new construction has a significant impact in practice.
2. MATRIX REPRESENTATION OF TRANSVERSAL DESIGNS
A pooling design is usually represented by a binary matrix with rows indexed with items and columns indexed with pools. A cell (i, j ) contains a 1-entry if and only if the ith pool contains the j th item. This binary matrix is called the incidence matrix of the represented pooling design. By treating a column as a set of row indices each intersecting the column with a 1-entry, we can talk about the union of several columns. A binary matrix is d-separable if every two unions from different subsets of d columns are different, is ¯d-separable if every two unions from different subsets of at most d columns are different, and is d-disjunct if no column is contained in a union of other d columns.
A transversal design has a special matrix representation with rows indexed by families and columns indexed by items; a cell (i, j ) contains entry k if and only if item j belongs to the kth pool in the ith family. This matrix representation is called a transversal matrix of the represented transversal design.
Each f × n matrix can be seen as a transversal matrix of a transversal design as follows: Use entries on the ith row to index pools in the ith family. The pool with index k in the ith family contains the j th item if and only if cell (i, j ) contains entry k in the matrix. For example, matrix
⎛
⎝12 13 −1 −12 3
0 1 1 2
⎞ ⎠
represents transversal design
{1, 2}, {3, 4}; {1, 3}, {2, 4};
{1}, {2, 3}, {4}.
In this way, each matrix represents a transversal design and each transversal design has more than one transversal matrix.
We can also extend the concept of d-separability, ¯d-separability, and d-disjunctness to the general matrix. For a general matrix, the union of d column vectors is defined to be a column vector each of whose components is the union of corresponding components of those d column vectors. A general matrix is d∗-separable ( ¯d∗-separable) if all unions of (at most) d columns are different. A general matrix is d∗-disjunct if no column is contained in the union of d other columns (i.e., at least one component of the column is not contained in the corresponding component of the union). For example, the reader may verify that the 3× 4 matrix in above example is 2∗-disjunct.
Theorem 1. A transversal design is d-separable if and only if its general matrix representation is d∗-separable. A transversal design is ¯d-separable if and only if its general matrix representation is ¯d∗ -separable. A transversal design is d-disjunct if and only if its general matrix representation is d∗-disjunct.
Proof. One may transform each general matrix representation of a transversal design to its binary matrix representation by replacing each row Ri by several rows with indices each being a pair of i and
an entry k of Ri and in row {i, k}, the cell ({i, k}, j) contains a 1-entry if and only if in the general
matrix representation, the cell (i, j ) contains entry k. Then it is easy to verify that the resulting binary matrix is d-separable if and only if the original general matrix is d∗-separable, the resulting binary matrix
is ¯d-seperable if and only if the original general matrix is ¯d∗-separable, and the resulting binary matrix is d-disjunct if and only if the original general matrix is d∗-disjunct.
It may be worth mentioning that a d-separable binary matrix must be d∗-separable, but a d∗-separable binary matrix may not be d-separable. Similar relations hold between ¯d-separability and ¯d∗-separability, and between d-disjunctness and d∗-disjunctness. This is why we use d∗instead of d in the terminologies involving general matrices.
3. A NEW CONSTRUCTION
We present a new construction of transversal design in this section. Consider a finite field GF (q) of order q. Suppose k satisfies
n≤ qk (1)
and
f = d(k − 1) + 1 ≤ q. (2)
We construct an f × n matrix M(d, n, q, k) as follows: Its column indices are polynomials of degree k over the finite field GF (q). Its row indices are f distinct elements of GF (q). The cell (x, g) contains element g(x) of GF (q).
Theorem 2. M(d, n, q, k)is a d∗-disjunct matrix.
Proof. Suppose M(d, n, q, k) is not d∗-disjunct. Then it has a column g0 contained in the union of other d columns g1, . . . , gd. That is, for each row index xi, g0(xi)= gj(xi)for some j . Note that there
are d(k− 1) + 1 rows. Thus, there exists a gj (1 ≤ j ≤ d) such that g0(xi)= gj(xi)for at least k row
indices xi. It follows that g0= gj, a contradiction.
By (1) and (2), k and q should be chosen to satisfy
logqn≤ k ≤ q− 1
d + 1. (3)
There exists a positive integer k satisfying (3) if q satisfies
logqn≤ q− 1
d . (4)
That is, it is sufficient to choose q satisfying
nd≤ qq−1. (5)
Let q0be the smallest number q satisfying (5). Then, we have the following estimation on q0.
Lemma 3. q0= (1 + o(1)) dlog2n log2(dlog2n). Moreover, q0≤ 1 + 2d log2n log2(dlog2n) for nd≥ 24.
Proof. Set
q1= 1 + (1 + h(d, n))
dlog2n log2(dlog2n),
where
h(d, n)= log2log2(dlog2n)
log2(dlog2n)− log2log2(dlog2n). Note that h(d, n)≥ 0. Therefore,
(q1− 1) log2q1> (q1− 1) log2(q1− 1) ≥ (1+ h(d, n))d log2n
log2(dlog2n) · log2
(1+ h(d, n))d log2n log2(dlog2n)
> dlog2n.
That is, q1satisfies (5). It follows that q0≤ q1. Note that h(d, n)= o(1). Hence,
q0= (1 + o(1))
dlog2n log2(dlog2n).
Moreover, for nd ≥ 24, d log2n ≥ 4. Hence, 2dlog2n ≥ (d log2n)2. Thus, d log
2n ≥ 2 log2(dlog2n). It follows that h(d, n)≤ 1. Therefore,
q0≤ 1 +
2d log2n log2(dlog2n)
for nd ≥ 24.
We need to find a prime power q satisfying
q ≥ q0. Then, we can choose
k= logqn. For such a choice of k, we have
f = d(k − 1) + 1 ≤ d(logqn − 1) + 1 ≤ d(logq0n − 1) + 1 ≤ q0. Since each family contains at most q pools, the total number of tests is at most q0q.
Theorem 4. There exist a prime power q and a positive integer k satisfying (1) and (2), such that M(d, n, q, k)gives a transversal design with at most 2q02 tests.
Proof. Set q = 2log2q0. Then q is a prime power satisfying q
0≤ q < 2q0). Therefore, qq0<2q02.
Corollary 5. There exists a transversal design M(d, n, q, k) with at most
(2+ o(1)) dlog2n log2(dlog2n) 2 tests.
There exist two previous constructions for transversal designs in the literature. The first one is the grid design (Barillot et al., 1991; Hwang, 1995; Phatarfod and Sudbury, 1994). With a k-dimensional grid, the number of tests can be O(dn1/k). The second is the Chinese remainder sieve (Eppstein et al., 2004) which uses O(ln(2d ln n)(dln n)2 ) tests. Our new construction uses O(( dlog2n
log2(dlog2n))
2) tests, which is better than both previous ones.
Moreover, this new construction can be easily extended to the error-tolerant case. Let e be the upper bound for the number of possible errors in testing. To have an error-tolerant property, a pooling design has to meet some stronger requirement. A pooling design is d#e-disjunct if its binary representation matrix satisfies the property that every column has at least e+ 1 1-entries not contained in the union of some other d columns.
Lemma 6. A transversal design is d#e-disjunct if and only if its general matrix representation has the property that every column has at least e+ 1 components not contained in the union of some other d columns.
Proof. Similar to the proof of Theorem 1.
Now, let us assume that q and k satisfy
n≤ qk (6)
and
f = d(k − 1) + 1 + e ≤ q. (7)
We construct an f × n matrix M(d, n, q, k, e) as follows: Its column indices are polynomials of degree k over the finite field GF (q). Its row indices are f distict elements of GF (q). The cell (x, g) contains element g(x) of GF (q).
Theorem 7. M(d, n, q, k, e) is a general matrix representation of d#e-disjunct transversal design.
Proof. Suppose M(d, n, q, k, e) is not d∗-disjunct. Then it has a column g0which has at least f − e components contained in the union of the other d columns g1, . . . , gd. Thus, there exists a column gj
containing at least k components of g0. That is, for at least k row indices xi, g0(xi)= gj(xi). Therefore,
g0= gj, a contradiction.
By an argument similar to the above, we can also obtain the following.
Theorem 8. By properly choosing q and k, we can obtain an M(d, n, q, k, e) with at most 2qe2 tests where qe= e + (2 + o(1)) 2d log2n log2(dlog2n) 2 .
4. DISCUSSION
The coefficient 2 in Theorems 7 and 8 can be further improved if we have better knowledge of the distribution of prime powers. In fact, if Goldbach’s conjecture about even numbers is true, that is, every even number is a sum of two primes, then there exists a prime between m and 2m for every natural number m. This means that there exist many prime powers between m and 2m. What is the smallest constant c such that there exists a prime power between m and cm? It is unknown and possibly an interesting open problem. It may have a relation to the number of representations being a sum of two prime powers for an even number.
ACKNOWLEDGMENTS
W.W. was supported in part by NSF grant ACI-0305567 and T.Z. was supported in part by NSF grant CCF-0548895.
REFERENCES
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Address correspondence to: Ding-Zhu Du Dept. of Computer Science University of Texas at Dallas MS EC31 2601 North Floyd Road Richardson, TX 75083
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