Abstract
We study group-testing algorithms for resolving broadcast conflicts on a multiple access channel (MAC) and for identifying the dead sensors in a mobile ad hoc wireless network. In group-testing algorithms, we are asked to identify all the defective items in a set of items when we can test arbitrary subsets of items. In the standard group-testing problem, the result of a test is binary—the tested subset either contains defective items or not. In the more generalized versions we study in this paper, the result of each test is non-binary. For example, it may indicate whether the number of defective items contained in the tested subset is zero, one, or at least two.
We give adaptive algorithms that are provably more efficient than previous group testing algorithms. We also show how our algorithms can be applied to solve conflict resolution on a MAC and dead sensor diagnosis. Dead sensor diagnosis poses an interesting challenge compared to MAC resolution, because dead sensors are not locally detectable, nor are they themselves active participants.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allemann A (2003) Improved upper bounds for several variants of group testing. PhD dissertation, Rheinisch-Westfalischen Technischen Hochschule Aachen
Atallah MJ, Goodrich MT, Tamassia R (2005) Indexing information for data forensics. In: 3rd applied cryptography and network security conference (ACNS). Lecture notes in computer science, vol 3531. Springer, Berlin, pp 206–221
Berger T, Mehravari N, Towsley D, Wolf J (1984) Random multiple-access communication and group testing. IEEE Trans Commun 32(7):769–779
Capetanakis JI (1979) Tree algorithms for packet broadcast channels. IEEE Trans Inf Theory 25(5):505–515
Christen CA (1994) Search problems: one, two or many rounds. Discret Math 136:39–51
Colbourn, Dinitz, Stinson (1999) Applications of combinatorial designs to communications, cryptography, and networking. In: Walker (ed) Surveys in combinatorics, 1993. London mathematical society iecture note series, vol 187. Cambridge University Press, Cambridge. Available at: citeseer.ist.psu.edu/colbourn99applications.html
DeBonis A, Gasieniec L, Vaccaro U (2003) Generalized framework for selectors with applications in optimal group testing. In: Proceedings of 30th international colloquium on automata, languages and programming (ICALP’03). Springer, Berlin, pp 81–96
Du D-Z, Hwang FK (2000) Combinatorial group testing and its applications, 2nd edn. World Scientific, Singapore
Eppstein D, Goodrich MT, Hirschberg DS (2007) Improved combinatorial group testing algorithms for real-world problem sizes. SIAM J Comput 36(5):1360–1375
Farach M, Kannan S, Knill E, Muthukrishnan S (1997) Group testing problems with sequences in experimental molecular biology. In: Sequences, IEEE Press, New York, p 357
Gargano L, Montuori V, Setaro G, Vaccaro U (1992) An improved algorithm for quantitative group testing. Discret Appl Math 36:299–306
Greenberg AG, Ladner RE (1983) Estimating the multiplicities of conflicts in multiple access channels. In: Proc. 24th annual symposium on foundations of computer science (FOCS’83). IEEE Computer Society, Los Alamitos, pp 383–392
Greenberg AG, Winograd S (1985) A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels. J ACM 32(3):589–596
Hofri M (1984) Stack algorithms for collision-detecting channels and their analysis: a limited survey. In: Balakrishnan AV, Thoma M (eds) Proceedings of the information seminar on modelling and performance evaluation methodology. Lecture notes in control and information sciences, vol 60, pp 71–85
Hwang FK (1972) A method for detecting all defective members in a population by group testing. J Am Stat Assoc 67:605–608
Hwang FK, Sós VT (1987) Non-adaptive hypergeometric group testing. Studia Sci Math Hung 22:257–263
Hwang FK, Song TT, Du DZ (1981) Hypergeometric and generalized hypergeometric group testing. SIAM J Algebr Discret Methods 2(4):426–428
Intanagonwiwat C, Estrin D, Govindan R, Heidemann J (2002) Impact of network density on data aggregation in wireless sensor networks. In: Proceedings of international conference on distributed computing systems
Kautz WH, Singleton RC (1964) Nonrandom binary superimposed codes. IEEE Trans Inf Theory 10:363–377
Lynch NA (1996) Distributed algorithms. San Francisco, Kaufmann
Macula AJ, Reuter GR (1998) Simplified searching for two defects. J Stat Plan Inf 66:77–82
Motwani R, Raghavan P (1995) Randomized algorithms. Cambridge University Press, New York
Ruszinkó M (1994) On the upper bound of the size of the r-cover-free families. J Comb Theory Ser A 66:302–310
Schlaghoff J, Triesch E (1997) Improved results for competitive group testing. Forschungsinstitut fur Diskrete Mathematik, Institut fur Okonometrie und Operations Research, Rheinische Friedrich-Wilhelms-Universitat Bonn, Report 97858
Schmidt JP, Siegel A, Srinivasan A (1993) Chernoff–Hoeffding bounds for applications with limited independence. In: ACM-SIAM symposium on discrete algorithms (SODA). Available at: citeseer.ist.psu.edu/19779.html
Yuan W, Krishnamurthy SV, Tripathi SK (2004) Improving the reliability of event reports in wireless sensor networks. In: Proceedings of IEEE international symposium on computers and communication (ISCC), pp 220–225
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper was presented at SPAA 2006.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10878-007-9087-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-007-9087-z