Threshold Group T esting with Consecutive Positives

1. Threshold Group Testing with Consecutive Positives Advisor : Huilan Chang Student : Yi-Lin Tsai Department of Applied MathematicsNational University of Kaohsiung 2014/08/02

2. Outline • Introduction • Group testing • Group testing with consecutive positives • Threshold group testing • Main result • Sequential algorithm for T.G.T.C • Nonadaptive algorithm for T.G.T.C Concluding Reference 2

3. Classical group testing • Given a set of items, each is either positive or negative, and a set of at most positives. • Goal : identify all positives by group test. • GroupTest : a test on a subset . negative positive Positive outcome: contains at least one positive item. 3

4. Types of algorithm • Sequential algorithm : A test can be specified after the previous test outcome. • Nonadaptive algorithm : All test are specified beforehand and are conducted simultaneously. items tests 4

5. Types of algorithm • Sequential algorithm : A test can be specified after the previous test outcome. • Nonadaptive algorithm : All test are specified beforehand and are conducted simultaneously. items 1 0 Outcome vector 0 1 0 0 tests 4

6. Consecutive model • is a set of items with the linear order for . • : is a set of positive items which is consecutive (under the ordering ), and contains at most items. • Test : choose arbitrary subset of . 5

7. Consecutive model • Balding and Torney (1997) and Colbourn (1999) first studied this model. • Colbourn (1999) • Mllerand Jimbo (2004) • Juan and Chang (2008) sequential : nonadaptive : Lower bound : nonadaptive : sequential : , for 6

8. Threshold group testing • Peter Damaschke (2006) arbitrary answer negative positive lower threshold upper threshold 7

9. Threshold group testing • Peter Damaschke (2006) • If then we can find all positives.If then we can only find a -approximate set. lower threshold upper threshold 7

10. Threshold group testing • A set is called -approximate if and . EX1Let .EX2 The classical group testing is the case of. a b -approximate set c d e 8

11. Our work Threshold group testing Group testing with consecutive positives Threshold group testing with consecutive positives 9

12. Our work • Threshold group testing with consecutive positives • Lower bound. • Sequential algorithm. • Nonadaptive algorithm and decoding complexity : . 10

13. Main result Sequential algorithm Nonadaptive algorithm

14. Sequential algo. for T.G.T.C Recall : at most positives … n items Itis usually assumed that . 12

15. Sequential algo. for T.G.T.C Information-theoretic lower bound : Proposition (Chang and Tsai, 2014) If , then the number of group tests required to identify all positive items from is at least . 13

16. Sequential algo. for T.G.T.C Our job : Provide an algorithm to locate all positive items from linear order and compare with the lower bound. at most positives … 14

17. Sequential algo. for T.G.T.C Our job : Provide an algorithm to locate all positive items from linear order and compare with the lower bound. at most positives … 14

18. Sequential algo. for T.G.T.C Our job : Provide an algorithm to locate all positive items from linear order and compare with the lower bound. at most positives … min max . We start with the case gap . . 14

19. Threshold without gap Theorem 1 (Chang and Tsai, 2014) For gap-free T.G.T.C, all positives can be identified in tests. 15

20. Threshold without gap Proof of Theorem 1 • First partition into parts of consecutive items andadd some dummy negative items to the last part. dummyitems • Let . • Goal : find min. Algorithm 1 and Algorithm 2 16

21. Threshold without gap Proof of Theorem 1 After Algorithm 1, 2, we have : min() Next, find max() : Apply a binary search algorithm to where each group test iscomposed of consecutive items. 17

22. Algorithm 1 FIND-TWO-CANDIDATES Positive : Negative : 18

23. Threshold without gap Proof of theorem 1 Lemma 1 (Chang and Tsai, 2014) FIND-TWO-CANDIDATES returns that minin tests. 19

24. Algorithm 2 LOCATE-STARTER 20

25. Algorithm 2 LOCATE-STARTER 20

26. Threshold without gap Proof of theorem 1 Lemma 2 (Chang and Tsai, 2014) LOCATE-STARTER can identify min() from in tests. 21

27. Threshold without gap Theorem 1 (Chang and Tsai, 2014) For gap-free T.G.T.C, all positives can be identified in tests. 22

28. Threshold with gap Theorem 2(Chang and Tsai, 2014) For T.G.T.Cwith , a -consecutive-approximateset can beidentified in tests. 23

29. Main result Sequential algorithm Nonadaptive algorithm

30. Nonadaptive algo. for T.G.T.C Recall : 1 0 0 1 0 0 25

31. Consecutive-disjunct matrix Definition 1 (Chang, Chiu and Tsai, 2014) A binary matrix is -consecutive-disjunct if for any cyclically consecutive columns and other cyclically consecutive columns , there exists one row intersecting but none of . 1111111 000000000 the minimum number of rows among of all -consecutive-disjunctmatrices of columns. 26

32. Consecutive-disjunct matrix • Probabilistic methodLovsz Local Lemma (1974) • Greedy constructionLovsz-Stein Theorem (1975) 27

33. Probabilistic method Lemma 3 (Lovsz Local Lemma) event. For each is dependent of at most events. for all . If ,then . 28

34. Probabilistic method Theorem 3(Chang, Chiu and Tsai, 2014) with and , Example. 29

35. Greedy construction Theorem 7 (Chang, Chiu and Tsai, 2014) with and where Example. 30

36. Nonadaptivealgo. for T.G.T.C Goal : Identify a -approximate set. 31

37. Nonadaptivealgo. for T.G.T.C Apply a -consecutive-disjunct matrix with columns. Given 32

38. Nonadaptivealgo. for T.G.T.C Theorem 8(Chang, Chiu and Tsai, 2014) For T.G.T.Cwith , nonadaptive algorithm can identify a -approximate set in tests. Furthermore, the decoding complexity is . Proof. . 33

39. Nonadaptivealgo. for T.G.T.C Theorem 9(Chang, Chiu and Tsai, 2014) For G.T.C, nonadaptive algorithm can identify all positives in tests. Furthermore, the decoding complexity is . Proof. . 34

40. Concluding • Threshold group testing with consecutive positives • Lower bound. • Sequential algorithm. • Nonadaptive algorithm and decoding complexity : . 35

41. References D.J.Balding and D. C. Torney, The design of pooling experimentsfor screening a clone map, Fungal Genet. Biol. 21 (1997) 302-307. H. Chang, Y.-C Chiu and Y.-L Tsai, A variation of cover-free families and its applications, preprint. H. Chang and Y.-L Tsai, Threshold group testing with consecutivepositives, Discrete Appl. Math. 169 (2014) 68-72. C. J. Colbourn, Group testing for consecutive positives, Ann. Combin. 3 (1999) 37-41. P.Damaschke, Threshold group testing, In: General Theory ofInformation Transfer and Combinatorics, Lect. Notes Comput. Sci. 4123 (2006) 707-718. P. Erdos and L. Lovasz, Infinite and finite sets, Colloq. Math. Soc. Janos Bolyai 10 (1974) 609-627. J. S.-T. Juan and G. J. Chang, Adaptive group testing for consecutivepositives, Discrete Math. 308 (2008) 1124-1129. 36

42. References L. Lovasz, On the ratio of optimal integral and fractional covers,Discrete Math. 13 (1975) 383-390. R. A. Moser and G. Tardos, A constructive proof of the general Lovasz Local Lemma, Journal of the ACM (JACM). 57 (2010) 1-15. M. Muller and M. Jimbo, Consecutive positive detectable matricesand group testing for consecutive positives, Discrete Math. 279(2004) 369-381. S. K. Stein, Two combinatorial covering problems, J. CombinatorialTheory. 16 (1974) 391-397. 37

43. Thank you for your attention!