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Principles of Group Testing and an Application to the Design and Analysis of Multi-Access Protocols

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The Impact of Processing Techniques on Communications

Part of the book series: NATO ASI Series ((NSSE,volume 91))

Abstract

In group testing we are concerned with determining the state of items, each one of which can be either in a good state (denoted “g”) or a defective state (denoted “d”). We assume that we have a universal test which can be applied simultaneously to any number of the items. In classical group testing there can be two outcomes of the universal test. One of the outcomes is that all items tested are good (denoted “G”) and the other outcome is that there was at least one defective in the group tested (denoted “D”). The choice of the items to be tested in any given to test can depend upon the entire history of the testing to that point. That is, the items to be tested in the i-th test can depend upon which items were tested in all preceding tests and the outcomes of these tests. A test plan is a set of test and the corresponding rules for choosing which items are to be tested in each test. If the number of items is finite, the test plan is concluded when the states of all items are determined. The problem of group testing is to specify an efficient test plan. Our criterion for efficiency will be discussed later.

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References

  1. R. Dorfman, “The Detection of Defective Members of Large Populations,” Annals of Mathematical Statistics, Vol. 14, pp. 436–440, 1943.

    Article  Google Scholar 

  2. A. Sterrett, “On the Detection of Defective Members of Large Populations,” Annals of Mathematical Statistics, Vol. 28, pp. 1033–1036, 1957.

    Article  MATH  Google Scholar 

  3. M. Sobel and P.A. Groll, “Group Testing to Eliminate Efficiently All Defectives in a Binomial Sample,” Bell Systems Technical Journal, Vol. 38, pp. 1178–1252, 1959.

    MathSciNet  Google Scholar 

  4. C.H. Li, “A Sequential Method for Screening Experimental Variables,” Journal American Statistical Association, Vol. 57, pp. 455–477 1962.

    Article  MATH  Google Scholar 

  5. A. Gill and D. Gottleib, “The Identification of a Set of Successive Interactions,” Information and Control, Vol. 24, pp. 20–35, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  6. G.T. Duncan, “Heterogeneous Questionnaire Theory,” SIAM Journal of Applied Mathematics, Vol. 27, pp. 59–71, 1974.

    Article  MATH  Google Scholar 

  7. M.R. Garey, ‘Optimal Binary Identification Procedures,’ SIAM Journal of Applied Mathematics, Vol. 23, pp. 173–186, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Sobel, “Group Testing to Classify Efficiently All Units in a Binomial Sample,” Information and Decision Processes, edited by Robert Machol, McGraw-Hill, New York, pp. 127–161, 1960.

    Google Scholar 

  9. M. Sobel, “Optimal Group Testing,” Colloquium on Information Theory, Bolyai Mathematics Society Debrecen, Hungary, 1967.

    Google Scholar 

  10. F.K. Hwang, “On Finding a Single Defective in Binomial Group Testing,” Journal of the American Statistical Association Theory and Methods Section, Vol. 69, pp. 146–150, 1974.

    Article  MATH  Google Scholar 

  11. F.K. Hwang, “On Optimum Nested Procedures in Binomial Group Testing,” Biometrics, Vol. 32, pp. 939–943.

    Google Scholar 

  12. F.K. Hwang, “A Generalized Binomial Group Testing Problem, Journal of the American Statistical Association Theory and Methods Section, Vol. 70, pp. 923–926,1975.

    Article  MATH  Google Scholar 

  13. F.K. Hwang, “Hypergeometric Group Testing Procedures and Merging Procedures,” Bulletin of the Institute of Mathematics, Academy Senica, Vol. 5, pp. 335–343, 1977.

    MATH  Google Scholar 

  14. F.K. Hwang, “A Note on Hypergeometric Group Testing Procedures,” SIAM Journal of Applied Mathematics, Vol. 34, pp. 371–375, 1978.

    MATH  Google Scholar 

  15. P. Ungar, “The Cutoff Point for Group Testing,” Communications on Pure and Applied Mathematics, Vol. XIII, pp. 49–54, 1960.

    Article  MathSciNet  Google Scholar 

  16. F.K. Hwang, S. Lin and C.L. Mallows, “Some Realizability Theorems in Group Testing, ” SIAM Journal on Applied Mathematics, Vol. 37, pp. 396–400, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  17. D.A. Huffman, “A Method for the Construction of Minimum Redundancy Codes,” Proceeding of the Institute of Radio Engineers, Vol. 40, p. 1098, 1952.

    Google Scholar 

  18. C.E. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal, Vol. 27, pp. 379–423 623–656, 1948.

    MathSciNet  MATH  Google Scholar 

  19. D. Towsley and J.K. Wolf, Group Testing, Polling, and Multi-Access Communications,” submitted for publication to IEEE Transactions on Communications.

    Google Scholar 

  20. J. Massey and J.K. Wolf, unpublished research conducted in August 1982 at the E.T.H., Zurich, Switzerland.

    Google Scholar 

  21. R.G. Gallager and D.C. Van Voorhis, “Optimal Source Codes for Geometrically Distributed Integer Alphabets,” IEEE Trans. on Information Theory, Vol. 21, pp. 228–229, 1975.

    Article  MATH  Google Scholar 

  22. M. Schwartz, Computer-Communication Network Design and Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1977: Chapters 1, 12, and 13.

    Google Scholar 

  23. L. Kleinrock, Queueing Systems, Vol. II: Computer Applications, John Wiley & Sons, New York, 1976: Chapters 5 and 6.

    Google Scholar 

  24. A. Tanenbaum, Computer Networks, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1981: Chapters 6 and 7.

    Google Scholar 

  25. N. Abramson, “Packet-Switching with Satellites,” AFIPS Conference Proceedings, National Computer Conference, Vol. 42, pp. 695–702, 1973.

    Google Scholar 

  26. R.G. Gallager, “Conflict Resolution in Random Access Broadcast Network,” Proc. AFOSR Workshop on Communication Theory and Application, pp. 74–76, 1978.

    Google Scholar 

  27. N. Pippenger, “Bounds on the Performance of Protocols for a Multiple-Access Broadcast Channel,” IEEE Transactions on Information Theory, Vol. IT-27, pp. 145–151, 1981.

    Article  MathSciNet  Google Scholar 

  28. J.I. Capetanakis, “Tree Algorithms for Packet Broadcast Channels,” IEEE Transactions on Information Theory, Vol. IT-25, pp. 505–515, 1979.

    Article  MathSciNet  Google Scholar 

  29. J. Mosely, An Efficient Contention Resolution Algorithm for Multiple Access Channels, LIDS-TH-918, Laboratory for Information and Decision Systems, MIT, Cambridge, MA, 1979.

    Google Scholar 

  30. M.L. Molle, “On the Capacity of Infinite Population Multiple Access Protocols,” IEEE Transactions on Information Theory, Vol. IT-28, pp. 396–401, 1982.

    Article  MathSciNet  Google Scholar 

  31. T. Berger, letter to J. Hayes, dated February 1, 1983.

    Google Scholar 

  32. N. Pippenger, private conversation with Toby Berger, February 1977.

    Google Scholar 

  33. T. Berger, proposal to National Science Foundation, 1979.

    Google Scholar 

  34. D. Towsley and J.K. Wolf, “An Application of Group Testing to the Design of Multi-User Access Protocols,” Proceedings of Nineteenth Annual Allerton Conference on Communications, Control and Computing, Urbana, Illinois, PP. 397–403, 1981.

    Google Scholar 

  35. B.S. Tsybakov and V. Mikhailov, “An Upper Bound to Capacity of Random Multiple Access Systems,” Probl. Peredach. Inform, Vol. 17, January, 1981.

    Google Scholar 

  36. R. Cruz and B. Hajek, “A New Upper Bound to the Throughput of a Multi-Access Broadcast Channel,” IEEE Trans. on Information Theory, Vol. IT-28,pp, 402–405, 1982.

    Article  Google Scholar 

  37. R. Rueppel, “A Unification of All Existing Upper Bounds for the Random-Access Problem,” IEEE International Symposium on Information Theory, St. Jovite, Quebec, Canada, September 26–30, 1983.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, Massachusetts, 01003, USA

    Jack Keil Wolf

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  1. Jack Keil Wolf
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Editor information

Editors and Affiliations

  1. Marconi Research Centre, GEC Research Laboratories, Great Baddow, Essex, UK

    J. K. Skwirzynski

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© 1985 Martinus Nijhoff Publishers, Dordrecht

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Wolf, J.K. (1985). Principles of Group Testing and an Application to the Design and Analysis of Multi-Access Protocols. In: Skwirzynski, J.K. (eds) The Impact of Processing Techniques on Communications. NATO ASI Series, vol 91. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5113-6_12

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  • DOI: https://doi.org/10.1007/978-94-009-5113-6_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8760-5

  • Online ISBN: 978-94-009-5113-6

  • eBook Packages: Springer Book Archive

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