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

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

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

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  1. Error-correcting Pooling DesignsandGroup Testing for Consecutive Positives Advisor：Huilan Chang Student：Yi-Tsz Tsai Department of Applied Mathematics National Kaohsiung University

  2. Outline • Error-correcting pooling designs • Constructed from vectors • Constructed from distance-regular graph • Two-stage algorithm for Group testing of consecutive positives 1

  3. Pooling designs 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 2

  4. Pooling designs • 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. 3

  5. Pooling designs Nonadaptive algorithm • Binary matrix representation： • Rows are tests. • Columns are items. • An entry if test contains item Testing outcome • 0 0 1 0 0 1 • 0 1 0 1 1 1 0 • 0 0 1 1 1 0 1 • 1 0 1 0 0 1 1 4

  6. Pooling designs 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. columns At least row 5

  7. Pooling designs 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 columns At least rows 6

  8. Pooling designs Nonadaptive algorithm • Note： • An -disjunct matrix is also called -disjunct. • An -disjunct matrix is fully-disjunctif it is not -disjunct whenever or . • Application：A -disjunct matrix is -error-correcting. 7

  9. Construction Error-correcting pooling designs Constructed from vectors Constructed from distance-regular graph 8

  10. Construction - sequences 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 . 9

  11. Construction - sequences 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 , 10

  12. Construction - sequences Theorem (D’ychakov et al., 05’) Let and . is fully -disjunct where . How about “for some ”？ 11

  13. Construction - sequences Definition 1 Let and . ：binary matrix by rows indexed and columns indexed by . For and , iff . • if and for otherwise. • Example： 12

  14. Construction - sequences Definition 1 Let and . ：binary matrix by rows indexed and columns indexed by . For and , iff . • if and for otherwise. • Example： 13

  15. Construction - sequences • Example：iff 14

  16. Construction - sequences Our result： Theorem 1 Theorem 2 Let and . Then is -disjunct, where . Let and . Then is -disjunct, where . 15

  17. Construction - sequences Analysis： designated -ary vector of weight -ary vector of weight Goal： How many thatand ? 16

  18. Construction - sequences Analysis： (Theorem 2) • Step1：Find the lower bound of the number of -ary vectors of weight satisfying and for each . {：and either or and } ( … … … …) ( … … … … ) ( … … … …) ( … … … …) ( … … … …) 17

  19. Construction - sequences Analysis： (Theorem 2) • Step1：Find the lower bound of the number of -ary vectors of weight satisfying and for each . {：and either or and } ( … … … …) ( … … … … ) ( … … … …) ( … … … …) ( … … … …) 17

  20. Construction - sequences Analysis： (Theorem 2) • Step1：Find the lower bound of the number of -ary vectors of weight satisfying and for each . {：and either or and } ( … … 0 0 … 00… 0) ( … … … …) ( … … … …) ( … … … …) ( … … … …) 17

  21. Construction - sequences Analysis： (Theorem 2) • Step2：Find • Fixed , choose satisfying wherever ( … … … …) ( … … 0 0 … 00… 0) ( … … … …) ( … … … …) ( … … … …) ( … … … …) 18

  22. Construction - sequences Analysis： (Theorem 2) • Step2：Find • Fixed , choose satisfying wherever ( … … … …) ( … … 0 0 … 00… 0) ( … … … …) ( … … … …) ( … … … …) ( … … … …) 18

  23. Construction - sequences Analysis： (Theorem 2) • Step2：Find • Fixed , choose satisfying wherever ( … … … 0…0) ( … … 0 0 … 00… 0) ( … … … …) ( … … … …) ( … … … …) ( … … … …) 18

  24. Construction - sequences Concluding 1： (1) (2) (3) D’ychakovet al. (2005)： by “containment relation” Our results： by “intersecting relation” For given and , if , then , where (2) or (3). 19

  25. Construction – Johnson graph 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. 20

  26. Construction – Johnson graph • -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： 123 124 456 125 • if 245 234 236 235 21

  27. Construction – Johnson graph • -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： 123 124 456 125 • if • is a -clique of size . 245 234 236 235 21

  28. Construction – Johnson graph 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 . 22

  29. Construction – Johnson graph Theorem (Bai et al., 09’) Let and . is fully -disjunct where . How about “iff”？ 23

  30. Construction – Johnson graph Definition 2 Let and . ：binary matrix with rows indexed by -clique with size and columns indexed by -clique with size such that iff . 24

  31. Construction – Johnson graph Theorem (Lvet al., 14’) Let and . Then is -disjunct, where . 25

  32. Construction – Johnson graph Our result： Theorem 3 Let and . Then is -disjunct, where . 26

  33. Construction – Johnson graph Concluding 2： • Lv et al. (14’) ： • Our result： Table 1：Some comparisons of error-tolerance capabilities of 27

  34. Conclusion-Pooling designs Our results for error-correcting capabilities： • Construction by intersecting relation： • -ary vectors • -cliques of Johnson graph 28

  35. Two-stage for CGT Two-stage algorithmfor group testing of consecutive positives 29

  36. Two-stage for CGT 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. 30

  37. Two-stage for CGT Group testing of consecutive positives • Nonadaptive algorithm： • Begin by partition into partsEach part contains consecutive items. • All positive items are contained in atmost two parts. 31

  38. Two-stage for CGT Group testing of consecutive positives • Nonadaptive algorithm： • Begin by partition into parts.Each part contains consecutive items. • All positive items are contained in atmost two parts. • Colburn (99’) ： Gray code tests. • Mller and Jimbo (04’)： consecutive positive detectable matrices tests. 32

  39. Two-stage for CGT 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. 33

  40. Two-stage for CGT -selector ： Definition (De Bonis et al., 05’) Given , a binary matrix is a -selectorif any submatrix of obtained by choosing out of arbitrary columns of contains at least distinct rows of the identity matrix . Arbitrary columns -selector 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 At least rows of 34

  41. Two-stage for CGT Trivial two-stage algorithm (De Bonis et al., 05’)： Theorem (De Bonis et al., 05’) • Partition into parts . Let , there exists a -selector of size , with Stage 1 Stage 2 -selector At most 3 parts left Identity 35

  42. Two-stage for CGT Trivial two-stage algorithm (De Bonis et al., 05’)： 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

  43. Two-stage for CGT -selector ： 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 . arbitrary consecutive • -selector • rows of (in the submatrix) 0 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 37

  44. Two-stage for CGT Our two-stage algorithm： • 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. Stage 1 Stage 2 At most ? parts left -selector Identity 38

  45. Two-stage for CGT Lemma After using a-selector, there remain at most 3 partsin Stage 1 and their union contains all positive items. • 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

  46. Two-stage for CGT Theorem 5( by Lovsz-Stein Theorem) Theorem 6 Let and , This trivial two-stage algorithm identifies all positives in group tests. Furthermore, the decoding complexity is . • In Stage 1,. 40

  47. Two-stage for CGT Concluding 3： Theorem 4 Theorem 6 The trivial two-stage algorithm which provided by De Bonis et al. identifies all positives in group tests. Furthermore, its decoding complexity is . By choosing a -selector in the first stage, the trivial two-stage algorithm identifies all positives in group tests. Furthermore, the decoding complexity is . 41

  48. Reference [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 -disjuntinclusion 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. Mllerand 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.

  49. Thank you for your attention.

  50. Pooling designs 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 columns At least rows 6

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