池設計和連接網路

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(1)行政院國家科學委員會專題研究計畫. 池設計和連接網路. 計畫類別：個別型計畫計畫編號： NSC92-2115-M-009-014執行期間： 92 年 08 月 01 日至 94 年 07 月 31 日執行單位：國立交通大學應用數學系(所). 計畫主持人：黃光明. 報告類型：精簡報告處理方式：本計畫可公開查詢. 中. 華. 民. 國 94 年 8 月 30 日. 成果報告.

(2) 中文摘要(1)： A construction of Pooling Designs with some Happy Surprises 一文是我和本系同仁翁志文與美籍和俄籍學者合作而成。此文已被 J. Computational Biology 雜誌接受(主編 Michael Waterman 是計算分子生物的開山祖師)，表示我們的研究成果已不僅是數學上的推論，也確有生物上的應用。此文討論如何用有限域的理論構造 DNA 上可糾錯的測試，有很多意外的結果。中文摘要(2)： Equivalence of the one-rate model to the classical model on strictly nonblocking switching networks 一文已刊於離散數學中著有聲譽的 SIAM J. Disc. Math. 此文證明了不阻塞通訊網路中一個令人意外的結果，即一個簡單模型和一個很複雜的模型等價。.

(3) A Construction of Pooling Designs with Some Happy Surprises A. G. D’yachkov Frank K. Hwang∗ Antony J. Macula† Pavel A. Vilenkin Chih-wen Weng‡ December 9, 2003. Abstract The screening of data sets for ”positive data objects” is essential to modern technology. A (group) test that indicates whether or not a positive data object is in a specific subset or pool of the data set can greatly facilitate the identification of all the positive data objects. A collection of tested pools is called a pooling design. Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we use the (linear) subspace relation coupled with general concept of a ”containment matrix” to construct pooling designs with surprisingly high degrees of error-correction (detection.) Errorcorrecting pooling designs are important to biotechnical applications where error rates often are as high as 15%. What is also surprising is that the rank of the pooling design containment matrix is independent of the number of positive data objects in the data set. Keywords: pooling designs, error-correction. ∗. This research was partially supported by a Republic of China NSC grant 92-2115-M009-014. † Corresponding author. This research was partially supported by NSF-DMS 0107179. ‡ Corresponding author. This research was partially supported by a Republic of China NSC grant 91-2005-M-009-008.. 1.

(4) 1. Introduction. The screening of biological sets of objects, e.g., blood samples, cells, clones, macromolecules, is an essential but often laborious aspect of modern biotechnology. In a few instances, the screening of large libraries, e.g., peptide, cDNA, monoclonal antibody, for a relatively few number of positive objects has become a routine experimental procedure. See [2]. Similar approaches have also been proposed for contig sequencing [8], determination of exon boundaries in eukaryotic genes [17], detecting gene complex[18], micro-array quality control [3] and disease gene mapping [7]. Whenever the objective is to find ”needles in a haystack” a test indicating whether at least one needle is in a specific part of the haystack can greatly facilitate the isolation of the ”needles”. Such tests are called binary group tests and the general mathematical method behind the identification of the ”needles” using such tests is called group testing [6]. If we have a finite ground set or population containing elements that can be uniquely characterized as positive or negative, we refer to the collection of positive elements as the positive subset P . In the abstract group testing problem, P must be identified by performing 0, 1 tests on subsets of the population. One applied aim is to consider screening situations where we have a biological set of objects containing a relatively small number data points (e.g., clones) which have a measurable attribute or function that can characterize them as ”positive”. This subcollection is initially unknown to the experimenter and it is the object of the search. A group of biological objects taken from a larger set of objects is called a pool. A pool assay is a 0, 1 test to determine if at least one member of the pool is positive. The practical goal here is to determine a large portion of P from the pool assays. The collection of pools taken from a biological set of objects is called a pooling design. The following comes from [2]. ”Much of the current effort of the Human Genome project involves the screening of a large recombinant DNA libraries to isolate clones containing a particular DNA sequence.” ”This screening is important for disease-gene mapping and also for large-scale clone mapping.” ”More generally, efficient screening techniques can facilitate a broad range of basic and applied biological research.” 2.

(5) For example, using probes to screen DNA libraries of clones fits the group testing paradigm in the following way: The population is the DNA library which consists of thousands of separate recombinant DNA clones each of which represents some contiguous piece of a contiguous superpiece of DNA. A unique, identifiable, predetermined, and contiguous DNA subpiece is called a sequenced tagged site (STS). A clone is called positive for an STS if it contains that STS. A pool is a subset of the clones that are mixed together and tested by exposing the entire group to a chemical probe. A pool is labeled positive for an STS if the probe chemically indicates its presence. In other words, if the tests are error-free, then a pool is labeled positive for an STS if and only if that pool contains at least one clone that contains that STS. Generally because bioinformatic applications are often automated, parallel rather than sequential screening methods are generally preferred. See [6] for other screening cost factors. Long before the advent of bioinformatics, consideration of analogous factors in other testing, screening, or coding situations lead to the development of nonadaptive group testing. See [4]. In NGT, one must decide exactly which pools to test before any testing occurs. A NGT algorithm is sometimes referred to as a one-stage algorithm. A twostage algorithm is a nearly nonadaptive algorithm. In a trivial two-stage algorithm, all non-trivial pools occur in the first stage. After the first stage is complete, one has a set the candidate positives. In the second stage, each candidate positive is individually tested to see whether or not it is an actual positive. When screening biological sets errors almost always occur during the testing procedure. This paper addresses a new class of pooling designs that can cope with large numbers of errors.. 2. d-disjunct matrices as nonadaptive pooling design models. We will use the terminology of clone library screening for convenience. Suppose there are n clones including at most d positive ones (others are negative). A pooling design M can be represented by a binary incidence matrix where the columns represent clones, the rows represent tests, and mij = 1 if and only if clone j is contained in the subset of test i. 3.

(6) Suppose there are n clones including at most d positive ones (others are negative). A (group) test is applicable to an arbitrary subset of clones with two possible outcomes: a negative outcome indicates all clones in the subset are negative, and a positive outcome indicates otherwise. A pooling design is a specification of all tests so that they can be performed simultaneously with the goal to identify all positive clones with a small number of tests. A pooling design M can be represented by a binary incidence matrix where the columns represent clones, the rows represent tests, and mij = 1 if and only if clone j is contained in the subset of test i. Suppose M has t rows. Then the t outcomes can also be represented by a t-vector V = (v1 , · · · , vt )t , where vi = 1 if and only if the outcome of test i is positive (vi = 0 otherwise). Note that V is the boolean sum of the set of positive clones. Therefore it is convenient to view a column vector C as a subset S of the base set {1, 2, · · · , t}, where i ∈ S if and only if C has an 1-entry in row i. Then we can say that V is the union of the set of positive clones. M is called d-disjunct if no union of any d columns covers another column. A d-disjunct matrix not only identifies the up-to-d positive clones, but with a simple decoding. Namely, a clone is positive if and only if it (as a column) is contained by V. This is because a negative clone (column) has at least one row not covered by the union of the up-to-d positive clones; such a row then has a negative outcome which identifies the clone as negative. The notion of d-disjunctness was first raised by Kautz and Singleton[11] in the study of superimposed codes. It was also studied by Erd¨os, Frankl and F¨ uredi[5] under the name of d-cover-free family in extremal set theory. ddisjunct matrices have become the most important tool in the construction of deterministic pooling designs. Although many constructions have been proposed, the existence of d-disjunct matrices is still sparse. Macula [13] proposed a novel way of constructing d-disjunct matrices which uses the containment relation in a structure. More specifically, let [m] := {1, 2, · · · , m} be the base set. Then each ofµthe n ¶ columns is labeled m by a (distinct) k subset of [m], assuming n ≤ , and each of the k µ ¶ m rows is labeled by a (distinct) d-subset of [m], where d < k < m. d 4.

(7) mij = 1 if and only if the label of row i is contained in the label of column j. He proved that M is d-disjunct. Huang and Weng [9] generalized the construction to arbitrary atomic semi-lattice where the elements can be ranked. Again, label the columns by a subset of the rank k elements and label the rows by all rank d elements, d < k, then M is d-disjunct. Ngo and Du [16] further extended the construction to some geometric structures like simplicial complexes, and some graph properties like matchings. It is safe to say the ”containment matrix” method has opened a new door for constructing d-disjunct matrices from many mathematical structures. However, the basic result in all these constructions is invariably that, to obtain a d-disjunct matrix, use all rank d elements for rows. One practical problem with this type of construction is that a large n forces S to be large. Then the number of tests could be too large as there are too many rank d elements. This led Macula [15] to propose using the rank 2 elements for rows, regardless of the real d. He showed that while there is no guarantee to identify all positive clones, the probability of success is still satisfactory when d does not deviate too much from 2. Ngo and Du made a similar comment. In this paper, we show that the containment matrix which use rank r of elements for rows has the degree d of disjunctness, where r can be much less than d. In fact r can be any number from 1 to k − 1 (k is the lever for columns), while d ≤ q r for some constant q. This is the first happy surprise. Since we can choose r = 1, we also have better control on the number of tests.. 3. The error-correcting capability. Biological experiments are notorious for producing erroneous outcomes. Therefore it would be wise for pooling designs to allow some outcomes to be affected by errors. Macula[14] proposed the notion of de -disjunct to reflect the error-correcting capability of a d-disjunct matrix. A d-disjunct matrix is de -disjunct if a column has at least e + 1 1-entries not covered by the union of any other d columns. d0 -disjunct would then be the regular d-disjunct. 5.

(8) In [10] it was misclaimed that a de -disjunct matrix can correct e errors. The argument was that if we try all subsets E of up to e columns as the candidate set of errors and adjust the outcome set V to V ∪ E, then when E is the true error set, a positive clone C must be contained in V ∪ E. On the other hand, a negative clone C has at least e + 1 1-entries not covered by the set of up to d positive clones, i.e., C has at least e + 1 negative outcomes. At most e of them can be converted to positive by errors, thus at least one negative outcome is not covered by V. The problem of this argument is that we need to show that C has at least one negative outcome not covered by V ∪ E. The following is a counterexample. Example 3.1. d = 2, e = 1. Column 1 is the only positive clone while v3 is an error.     1 0 0 1  1 0 0   1       0 1 0   1      M = V =   0  0 1 0      0 0 1   0  0 0 1 0 When E = {4}, V ∪ E = (1, 1, 1, 1, 0, 0)t covers column 1 and 2. Thus the correct version should be Theorem 3.2. A d2e -disjunct matrix is e-error-correcting. Proof. For a positive clone C the argument is as before that there exists a candidate set E such that C ⊆ V ∪ E. A negative clone C has at least 2e + 1 1-entries not covered by the set D of up to d positive clones, hence at least 2e + 1 negative outcomes. e of them may be converted to positive by errors and another e of them by E, but at least one negative outcome is not covered by V ∪ E. For the reason that a de -disjunct matrix is not really e-correcting, and also that d0 -disjunct= d-disjunct, is kind of uncustomary, we propose to use the term dz -disjunct while z is the minimum (over C) number of 1-entries in C not covered by the union of any other d columns. Theorem 3.2 then can be restated as. 6.

(9) Theorem 3.3. A dz -disjunct matrix can detect z − 1 errors and correct z−1 b c errors. 2 In particular, a d-disjunct matrix has no error-tolerance. If an extra round of confirmatory tests is allowed, then a dz -disjunct matrix can indeed correct z − 1 errors. First, we need a lemma. Let H(X, Y ) denote the Hamming distance between two binary vectors X, Y of the same length. Lemma 3.4. Let M be a dz -disjunct matrix and let S1 S2 be two distinct subsets of columns with |S1 | ≤ d, |S2 | ≤ d. Let Ui be the union of the set Si for each i = 1, 2. Then 1. H(U1 , U2 ) ≥ z if either S1 ⊆ S2 or S2 ⊆ S1 ; 2. H(U1 , U2 ) ≥ 2z if otherwise. Proof. These are trivial by using the dz -disjunct property. Theorem 3.5. A dz -disjunct matrix corrects z −1 errors with an extra round of at most d confirmatory tests. Proof. Take all subsets S of columns of M with |S| ≤ d and H(U, V ) ≤ z −1, where U is the union of S. Let S1 , S2 be two such sets. The H(U1 , U2 ) ≤ 2(z − 1) < 2z. By Lemma 3.4, either S1 ⊆ S2 or S2 ⊆ S1 . Therefore the set {S} is a chain. Hence {S} has at most d members. Since H(D, V ) ≤ z − 1, D ∈ {S}, D can be identified by testing at most d columns in the maximal chain of {S}. Not many constructions of dz -disjunct matrices have been known. Macula [14], and also see [10], gave a construction for d4 , and recently Ngo and Du gave a construction for dd+1 . We will show that the construction delivering the first happy surprise mentioned in section 1 not only yields d-disjunct matrices, but also dz -disjunct matrices with the z-value much greater than 4 or d + 1. This is the second happy surprise.. 4. The construction. Consider the m-dimensional space, · or ¸ simply m-space, of GF (q) where q is m denote the number of k-dimensional a prime or a prime power. Let k q subspaces, or simply k-space. It is well known [12, p. 291] 7.

(10) Lemma 4.1.. ·. m k. ¸ q. (q m − 1)(q m−1 − 1) · · · (q m−k+1 − 1) = (q k − 1)(q k−1 − 1) · · · (q − 1). and. ·. m k. ¸. · = q. m m−k. ¸ . q. Definition 4.2. Fix integers 1 ≤ r < k < m. Let M (m, k, r) be the 01matrix by taking all k-spaces (from an underlying m-space) as columns and all r-spaces as rows. M (m, k, r) has a 1 in row i and column j if and only if i is contained in j. M (m, k, r) was first studied in [19] from a linear algebra point of view and in [16] from a pooling design point of view. M (m, k, r) is easily checked to be a ranked atomic semi-lattice, thus the matrix is r-disjunct, hence [9] dz -disjunct for some 1 ≤ d ≤ r and · ¸ k−d z= . r−d q Note that the construction still requires the row rank r being at least as large as the upper bound d of the number of positive clones. We now show that r can be much less than d. First, we give a lemma. Lemma 4.3. · ¸ · ¸ · ¸ k−1 k−1 k k−r − =q r r−1 q r q q Proof. ·. k r k. ¸. · − q. k−1 r. (0 ≤ r < k).. ¸ q. k−1. (q − 1)(q − 1) · · · (q k−r+1 − 1) (q k−1 − 1)(q k−2 − 1) · · · (q k−r − 1) − (q r − 1)(q r−1 − 1) · · · (q − 1) (q r − 1)(q r−1 − 1) · · · (q − 1) (q k − 1) − (q k−r − 1) (q k−1 − 1) · · · (q k−r+1 − 1) · = qr − 1 (q r−1 − 1) · · · (q − 1) · ¸ k−1 = q k−r . r−1 q =. 8.

(11) Theorem 4.4. Suppose k −r ≥ 2 and set p :=. q(q k−1 − 1) . Then M (m, k, r) q k−r − 1. is dz -disjunct for 1 ≤ d ≤ p and · ¸ · ¸ k−1 k−2 k−r k−r−1 z=q − (d − 1)q . r−1 q r−1 q. Proof. Let C, C1 , · · · , Cd· be ¸d + 1 distinct columns (k-spaces) of M . By k Lemma 3.1, C contains r-spaces. To obtain the maximum coverage r q of these r-spaces, we may assume that each Ci intersects C at a (k − 1)-space by the observation d d [ [ C∩ Ci = (C ∩ Ci ). ·. i=1. ¸. i=1. k−1 r-spaces of C. However, the coverage of r q each pair· of Ci and ¸ Cj overlaps at a (k − 2)-space. Therefore only C1 covers k−1 r-spaces, while each of C2 , · · · , Cd can cover a maximum the full r q ¸ · ¸ · k−2 k−1 − r-spaces not covered by C1 . Consequently the of r r q q number of r-spaces of C not covered by C1 , · · · , Cd is at least ¸ ¸ · ¸ · · · ¸ k−2 k−1 k−1 k ) − − (d − 1)( − z = r r r r q q q q ¸ · ¸ · k−2 k−1 k−r k−r−1 = q − (d − 1)q . r−1 q r−1 q. Then each Ci covers. Note that for M (m, k, r) to be dz -disjunct, z must be positive, which implies ¸ · k−1 k−r q r−1 q · ¸ + 1, d< k − 2 q k−r−1 r−1 q or d ≤ p. Suppose d ≤ q + 1. The following corollary shows the above z is optimal. 9.

(12) Corollary 4.5. Suppose k − r ≥ 2 and 1 ≤ d ≤ q + 1. Then M (m, k, r) is not dz+1 -disjunct, where z is as in Theorem 4.4. Proof. We prove this by showing that a maximum coverage of r-spaces in the proof of Theorem 4.4 is obtained. We reverse the arguments. Let U be a (k − 2)-space contained in C. Then the number of (k − 1)-spaces between U and C is · ¸ k − (k − 2) = q + 1. k − 1 − (k − 2) q We choose d distinct ones among them, say Ti (1 ≤ i ≤ d). For each Ti , we choose a k-space Ci such that C ∩ Ci = Ti . Hence each pair of Ci and Cj overlaps at the same (k − 2)-space U. k Lemma 4.6. Suppose r ≤ . Then with referring to the definition of p in 2 Theorem 4.4, d = q r is the largest integer less or equal to p. Proof. Note that q r−1 < q r ≤ q k−r . Hence p − qr = = = < ≤. q(q k−1 − 1) − qr q k−r − 1 qk − q − qk + qr q k−r − 1 q(q r−1 − 1) q k−r − 1 qq r−1 q k−r 1.. Then p − 1 < q r ≤ p. Corollary 4.7. Suppose k−r ≥ 2 and d = q r . Then M (m, k, r) is dz -disjunct with · ¸ · ¸ k−1 k−2 r z= + (q − 1) . r−1 q r q. 10.

(13) Proof. Setting d = q r in Theorem 4.4 and referring to Lemma 4.1, Lemma 4.3, · ¸ · ¸ k−1 k−2 k−r r k−r−1 z = q − (q − 1)q r−1 q r−1 q · ¸ · ¸ · ¸ · ¸ k k−1 k−1 k−2 r = − − (q − 1)( − ) r q r r r q q q · ¸ · ¸ · ¸ k k−1 k−2 r r = −q + (q − 1) r q r r q q · ¸ · ¸ · ¸ k k − 1 k−2 k−(k−r) r = −q + (q − 1) k−r q r r q q · ¸ · ¸ k−1 k−2 = + (q r − 1) k−r q r q ¸ · ¸ · k−2 k−1 + (q r − 1) . = r r−1 q q. When r = 1, the z in Theorem 4.4 is in a neater form. Corollary 4.8. Suppose k ≥ 3, d ≤ q and z = q k−2 (q − d + 1). Then M (m, k, 1) is dz -disjunct, but is not dz+1 -disjunct. Proof. Setting r = 1 in the z formula of Theorem 4.4, we obtain ¸ · ¸ · k−2 k−1 k−2 k−1 − (d − 1)q z = q 0 0 q q = q k−2 (q − d + 1). The second statement follows from Corollary 4.5. Example 4.9. Fix q = 5. Then M (8, 4, 1) is a 525 -disjunct matrix with 97656 rows and 200525284806 columns. This means that we can use around 105 pools with 25 errors allowed to determine the positives when the number of items is around 2 × 1011 with at most 5 positives.. 11.

(14) References [1] Barrllot, E., Lacroix, B. and Cohen, D. 1995. Theoretical analysis of library screening using an n-dimensional pooling strategy. Nucleic Acids Res. 19:6241-6247. [2] Bruno, W. J., Knill, E., Balding, D. J., Bruce, D. C., Doggett, N. A., Sawhill, W. W., Stallings, R. L., Whittaker C. C., and Torney E. C. 1995. Effective pooling designs of library screening. Genomics 26:21-30. [3] Colburn, C., Ling, A., Tompa, M. 2002. Construction of optimal quality control ogilio arrays, Bioinformatics, 18, no. 4, 529-535. [4] Du, D. and Hwang, F. K. 2000. Combinatorial Group Testing and Its Applications, 2nd Ed., World Scientific, Singapore. [5] Erd¨os, P., Frankl, and F¨ uredi, D. 1985. Families of finite sets in which no set is covered by the union of r others. Israel J. Math. 51:79–89. [6] Farach, et al. 1997. Group testing problems with sequences experimental molecular biology, Proceedings of Compression and Complexity of Sequences, 1997, B. Carpentieri, et al. (Eds.) IEEE Press, 357-367. [7] Flodman, P., Macula, A., Spence, A. and Torney, D. 2001. A new data mining technique for the analysis of simulated genetic data. Proceedings of Genetic Analysis Workshop 12, Wiley-Liss, 2001; Genet Epidemiol (Genetic epidemiology.) 21 Suppl 1: S390-5 (2001.) [8] Grebinski, V. and Kucherov, G. 1998. Reconstructing a Hamiltonian cycle by querying the graph:application to DNA physical mapping. Disc. Appli. Math.. 88:147-165. [9] Huang, T. and Weng, C. 2004. Pooling Spaces and Non-Adaptive Pooling Designs. Disc. Math. 282(1-3):163-169. [10] Hwang, F. K. 2003. On Macula’s Error-Correcting Pool Designs. Disc. Math. 267:311-314. [11] Kautz, W. H. and Singleton, R. C. 1964. Nonadaptive binary superimposed codes. IEEE Trans. Inform. Theory 10:363–377.. 12.

(15) [12] van Lint, J. H., and Wilson, R. M. 1992. A Course in Combinatorics. Cambridge, Victoria. [13] Macula, A. J. 1996. A simple construction of d-disjunct matrices with certain constant weights. Disc. Math. 162:311–312. [14] Macula, A. J. 1996. Error-correcting nonadaptive group testing with de -disjunct matrices. Disc. Appl. Math. 80:217-222. [15] Macula, A. J. 1999. Probabilistic nonadaptive and two-stage group testing with relatively small pools and DNA library screening. J. Comb. Opt. 2:385–397. [16] Ngo, H., and Du, D. 2002. New Constructions of Non-Adaptive and Error-Tolerance Pooling Designs. Disc. Math. 243:161–170. [17] Pevzner, P. A. 2000. Computational Molcular Biology: an algorithmic approach. MIT Press, Mass Sec. 9.5-9.6. [18] Torney, D. C. 1999. Sets pooling designs. Ann. Combin. 3:95–101. [19] Yakir, A. 1993. Inclusion matrix of k vs 1 affine subspaces and a permutation module of the general affine group. J. Combin. Theory, Ser. A 63:301–317.. A. G. D’yachkov Faculty of Mechanics and Mathematics Department of Probability Theory Moscow State University Moscow 119899, Russia email: [email protected] Frank K. Hwang Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road Hsinchu 30050, Taiwan 13.

(16) email: [email protected] Antony J. Macula* Department of Mathematics College at Geneseo State University of New York Geneseo, NU 14454, USA email: macula@geneseo Pavel A. Vilenkin Faculty of Mechanics and Mathematics Department of Probability Theory Moscow State University Moscow 119899, Russia email: [email protected] Chih-wen Weng Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road Hsinchu 30050, Taiwan email: [email protected]. 14.

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