The Journal of Korea Institute of Information, Electronics, and Communication Technology (한국정보전자통신기술학회논문지)

Korea Information Electronic Communication Technology (한국정보전자통신기술학회)
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A Lower Bound for Performance of Group Testing Problems

그룹검사 문제에 대한 성능 하한치

  • Seong, Jin-Taek (Department of Convergence Software, Mokpo National University)
  • Received : 2018.09.19
  • Accepted : 2018.10.15
  • Published : 2018.10.30
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Abstract

This paper considers Group Testing as one of combinatorial problems. The group testing first began to inspect soldier's syphilis infection during World War II and have long established an academic basis. Recently, there has been much interest in related areas because of the rediscovery of the value of the group testing. The group testing is the same as finding a few defect samples out of a large number of samples, which is similar to the inverse problem of Compressed Sensing. In this paper, we introduce the definition of the group testing, and specify the classes of the group testing and the bounds on performance of the group testing. In addition, we show a lower bound for the number of tests required to find defective samples using the theoretical theorem which is mainly used for relationship between conditional entropy and the probability of error in the information theory. We see how our result can be different from other related results.

본 논문은 조합 문제의 하나로써 그룹검사(Group Testing)의 성능 하한치를 유도한다. 그룹검사는 2차 세계대전 동안 군인들의 매독 감염을 검진하기 위해 시작되었고 지금까지 오랫동안 학문적 기초를 마련하였다. 최근 들어 그룹검사의 활용가치가 증대되어 재발견됨으로써 학계에서 큰 관심을 받고 있다. 그룹검사는 다수의 샘플 중에서 극소수의 결함 샘플을 찾는 문제와 동일하며, 이것은 압축센싱(Compressed Sensing)의 선형 역문제(inverse problem)와 유사하다. 본 논문에서는 그룹검사가 무엇인지 살펴보고 그룹검사의 관련 연구내용을 알아본다. 정보이론에서 사용한 조건부 엔트로피와 에러율 간의 관계를 밝히는 정리를 이용하여 결함 샘플을 찾기 위해 필요한 검사 수에 대한 최소 에러율의 경계값을 도출할 뿐만 아니라 기존 연구와 어떠한 차이점이 있는지 살펴본다.

Keywords

References

  1. Dorfman, Robert, "The Detection of Defective Members of Large Populations," The Annals of Mathematical Statistics, vol. 14, no. 4, pp. 436-440, Dec. 1943. https://doi.org/10.1214/aoms/1177731363
  2. D. L. Donoho, "Compressed Sensing," IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289-1306, Apr. 2006. https://doi.org/10.1109/TIT.2006.871582
  3. D.-Z. Du and F. K. Hwang, Pooling Designs and Nonadaptive Group Testing: Important Tools for DNA Sequencing, World Scientific, 2006.
  4. L. Baldassini, O. Johnson, M. Aldridge, "The capacity of adaptive group testing," IEEE International Symposium on Information Theory, pp. 2676-2680, Oct. 2013.
  5. A. Sterrett, "On the Detection of Defective Members of Large Populations," The Annals of Mathematical Statistics, vol. 28, no. 4, pp. 1033-1036, Dec. 1957. https://doi.org/10.1214/aoms/1177706807
  6. M. Sobel, P. A. Groll, "Group testing to eliminate efficiently all defectives in a binomial sample," Bell System Technical Journal, vol. 38, no. 5, pp. 1179-1252, Sep. 1959. https://doi.org/10.1002/j.1538-7305.1959.tb03914.x
  7. G. K. Atia and V. Saligrama, "Boolean Compressed Sensing and Noisy Group Testing," IEEE Transactions on Information Theory, vol. 58, no. 3, pp. 1880-1901. Mar. 2012. https://doi.org/10.1109/TIT.2011.2178156
  8. C. H. Li, "A Sequential Method for Screening Experimental Variables," Journal of the American Statistical Association, vol. 57, no. 298, pp. 455-477. Jun. 1962. https://doi.org/10.1080/01621459.1962.10480672
  9. J. N. Srivastava, A Survey of Combinatorial Theory, North-Holland, pp. 285-308, 1973.
  10. F. K. Hwang, "A Method for Detecting All Defective Members in a Population by Group Testing," Journal of the American Statistical Association, vol. 67, no. 339, pp. 605-608, Sep. 1972. https://doi.org/10.1080/01621459.1972.10481257
  11. A. Allemann, "An Efficient Algorithm for Combinatorial Group Testing," Information Theory, Combinatorics, and Search Theory, pp. 569-596, 2013.
  12. M. C. Hu, F. K. Hwang, and J. K. Wang, "A Boundary Problem for Group Testing," SIAM Journal on Algebraic Discrete Methods, vol. 2, no. 2, pp. 81-87, Jun. 1981. https://doi.org/10.1137/0602011
  13. L. Riccio and C. J. Colbourn, "Sharper bounds in adaptive group testing," Taiwanese Journal of Mathematics, vol. 4, no. 4, pp. 669-673, Jan. 2000. https://doi.org/10.11650/twjm/1500407300
  14. M.-G. Leu, "A note on the Hu-Hwang-aWang conjecture for group testing," The ANZIAM Journal, vol. 49, no. 4, pp. 561-571, Apr. 2008. https://doi.org/10.1017/S1446181108000175
  15. C. L. Chan, P. H. Che, S. Jaggi, and V. Saligrama, "Non-adaptive probabilistic group testing with noisy measurements: near-optimal bounds with efficient algorithms," 49th Annual Allerton Conference on Communication, Control, and Computing, pp. 1832-1839. Sep. 2011.
  16. M. Aldridge, L. Baldassini, and O. Johnson, "Group Testing Algorithms: Bounds and Simulations," IEEE Transactions on Information Theory, vol. 60, no. 6, pp. 3671-3687, Jun. 2014. https://doi.org/10.1109/TIT.2014.2314472