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A Toolbox for Provably Optimal Multistage Strict Group Testing Strategies

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Computing and Combinatorics (COCOON 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7936))

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Abstract

Group testing is the problem of identifying up to d defectives in a set of n elements by testing subsets for the presence of defectives. Let t(n,d,s) be the optimal number of tests needed by an s-stage strategy in the strict group testing model where the searcher must also verify that no more than d defectives are present. We develop combinatorial tools that are powerful enough to compute many exact t(n,d,s) values. This extends the work of Huang and Hwang (2001) for s = 1 to multistage strategies. The latter are interesting since it is known that asymptotically nearly optimal group testing is possible already in s = 2 stages. Besides other tools we generalize d-disjunct matrices to any candidate hypergraphs, which enables us to express optimal test numbers for s = 2 as chromatic numbers of certain conflict graphs. As a proof of concept we determine almost all test numbers for n ≤ 10, and t(n,2,2) for some larger n.

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References

  1. Balding, D.J., Torney, D.C.: Optimal Pooling Designs with Error Detection. J. Comb. Theory A 74, 131–140 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, H.B., Hwang, F.K.: Exploring the Missing Link Among d-Separable, \(\bar{d}\)-Separable and d-Disjunct Matrices. Discr. Appl. Math. 155, 662–664 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, Y., Du, D.Z.: New Constructions of One- and Two-Stage Pooling Designs. J. Comp. Biol. 15, 195–205 (2008)

    Article  MathSciNet  Google Scholar 

  4. Damaschke, P.: Optimal Randomized Group Testing: A Canonical Form and the One-Defective Case. In: Cicalese, F., Porat, E. (eds.) ICALP2011GT (informal proceedings), Zürich, pp. 55–67 (2011)

    Google Scholar 

  5. De Bonis, A., Di Crescenzo, G.: Combinatorial Group Testing for Corruption Localizing Hashing. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 579–591. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. De Bonis, A., Gasieniec, L., Vaccaro, U.: Optimal Two-Stage Algorithms for Group Testing Problems. SIAM J. Comp. 34, 1253–1270 (2005)

    Article  MATH  Google Scholar 

  7. Du, D.Z., Hwang, F.K.: Combinatorial Group Testing and Its Applications. Series on Appl. Math., vol. 18. World Scientific (2000)

    Google Scholar 

  8. Du, D.Z., Hwang, F.K.: Pooling Designs and Nonadaptive Group Testing. Series on Appl. Math., vol. 18. World Scientific (2006)

    Google Scholar 

  9. Dyachkov, A.G., Rykov, V.V.: Bounds on the Length of Disjunctive Codes. Problems of Info. Transmission 18, 7–13 (1982) (in Russian)

    MathSciNet  Google Scholar 

  10. Eppstein, D., Goodrich, M.T., Hirschberg, D.S.: Improved Combinatorial Group Testing Algorithms for Real-World Problem Sizes. SIAM J. Comp. 36, 1360–1375 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fang, J., Jiang, Z.L., Yiu, S.M., Hui, L.C.K.: An Efficient Scheme for Hard Disk Integrity Check in Digital Forensics by Hashing with Combinatorial Group Testing. Int. J. Digital Content Technol. and its Appl. 5, 300–308 (2011)

    Article  Google Scholar 

  12. Fischer, P., Klasner, N., Wegener, I.: On the Cut-off Point for Combinatorial Group Testing. Discr. Appl. Math. 91, 83–92 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, H., Hwang, F.K., Thai, M.T., Wu, W., Znati, T.: Construction of d(H)-Disjunct Matrix for Group Testing in Hypergraphs. J. Comb. Opt. 12, 297–301 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goodrich, M.T., Hirschberg, D.S.: Improved Adaptive Group Testing Algorithms with Applications to Multiple Access Channels and Dead Sensor Diagnosis. J. Comb. Optim. 15, 95–121 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, S.H., Hwang, F.K.: When is Individual Testing Optimal for Nonadaptive Group Testing? SIAM J. Discr. Math. 14, 540–548 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mézard, M., Toninelli, C.: Group Testing With Random Pools: Optimal Two-Stage Algorithms. IEEE Trans. Info. Th. 57, 1736–1745 (2011)

    Article  Google Scholar 

  17. Spencer, J.: Minimal Completely Separating Systems. J. Combin. Theory 8, 446–447 (1970)

    Article  MATH  Google Scholar 

  18. http://www.redcrossblood.org/learn-about-blood/what-happens-donated-blood/blood-testing (version as of January 2013)

  19. Xuan, Y., Shin, I., Thai, M.T., Znati, T.: Detecting Application Denial-of-Service Attacks: A Group-Testing-Based Approach. IEEE Trans. Par. Distr. Syst. 21, 1203–1216 (2010)

    Article  Google Scholar 

  20. Zhang, B.: Group Testing Regression Models. Dissertation, Dept. of Statistics, Univ. of Nebraska, Lincoln (2012)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Chalmers University, 41296, Göteborg, Sweden

    Peter Damaschke & Azam Sheikh Muhammad

Authors
  1. Peter Damaschke
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  2. Azam Sheikh Muhammad
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Editors and Affiliations

  1. Department of Computer Science, University of Texas at Dallas, 75080, Richardson, TX, USA

    Ding-Zhu Du

  2. College of Computer Science and Technology, Zhejiang University, 310027, Hangzhou, China

    Guochuan Zhang

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Damaschke, P., Muhammad, A.S. (2013). A Toolbox for Provably Optimal Multistage Strict Group Testing Strategies. In: Du, DZ., Zhang, G. (eds) Computing and Combinatorics. COCOON 2013. Lecture Notes in Computer Science, vol 7936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38768-5_40

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  • DOI: https://doi.org/10.1007/978-3-642-38768-5_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38767-8

  • Online ISBN: 978-3-642-38768-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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