• Gianluca De Marco
• Tomasz Jurdziński
• Michał Różański
• Grzegorz Stachowiak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)

## Abstract

We consider threshold group testing – a generalization of a well known and thoroughly examined problem of combinatorial group testing. In the classical setting, the goal is to identify a set of positive individuals in a population, by performing tests on pools of elements. The output of each test is an answer to the question: is there at least one positive element inside a query set Q ? The threshold group testing is a natural generalization of this classical setting which arises when the answer to a test is positive if at least $$t>0$$ elements under test are positive.

We show that there exists a testing strategy for the threshold group testing consisting of $$O(d^{3/2}\log (N/d))$$ tests, for d positive items in a population of size N. For any value of the threshold t, we also provide a lower bound of order $$\varOmega \left( \min \left\{ \left( \frac{d}{t}\right) ^2,\frac{N}{t}\right\} \right)$$. Our subquadratic bound shows a complexity separation with the classical group testing (which corresponds to $$t = 1$$) where $$\varOmega (d^2 \log _d N)$$ tests are needed [25].

Next, we introduce a further generalization, the multi-threshold group testing problem. In this setting, we have a set of $$s > 0$$ thresholds, $$t_1,t_2, \ldots , t_s$$. The output of each test is an integer between 0 and s which corresponds to which thresholds get passed by the number of positives in the queried pool. Here, one may be interested in minimizing not only the number of tests, but also the number of thresholds which is related to the accuracy of the tests. We show the existence of two strategies for this problem. The first one of size $$O(d^{3/2}\log (N/d))$$ is an extension of the above-mentioned result. The second strategy is more general and works for a range of parameters. As a consequence, we show that $$O(\frac{d^2}{t}\log (N/d))$$ tests are sufficient for $$t\le d/2$$. Both strategies use respectively $$O(\sqrt{d})$$ and $$O(\sqrt{t})$$ thresholds.

## Keywords

Group testing Threshold group testing Non-adaptive strategies Randomized algorithms

## Notes

### Acknowledgments

The authors would like to thank Darek Kowalski for his comments to the paper.

## References

1. 1.
Alon, N., Hod, R.: Optimal monotone encodings. IEEE Trans. Inf. Theory 55(3), 1343–1353 (2009)
2. 2.
Chen, H.-B., Fu, H.-L.: Nonadaptive algorithms for threshold group testing. Discrete Appl. Math. 157(7), 1581–1585 (2009)
3. 3.
Cheraghchi, M.: Improved constructions for non-adaptive threshold group testing. Algorithmica 67(3), 384–417 (2013)
4. 4.
Chlebus, B.S., De Marco, G., Kowalski, D.R.: Scalable wake-up of multi-channel single-hop radio networks. Theoret. Comput. Sci. 615, 23–44 (2016)
5. 5.
Chlebus, B.S., De Marco, G., Kowalski, D.R.: Scalable wake-up of multi-channel single-hop radio networks. In: Aguilera, M.K., Querzoni, L., Shapiro, M. (eds.) OPODIS 2014. LNCS, vol. 8878, pp. 186–201. Springer, Cham (2014). doi: Google Scholar
6. 6.
Chlebus, B.S., De Marco, G., Talo, M.: Naming a channel with beeps. Fundam. Inf. 153(3), 199–219 (2017)
7. 7.
Chlebus, B.S., Kowalski, D.R.: Almost optimal explicit selectors. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 270–280. Springer, Heidelberg (2005). doi:
8. 8.
Chrobak, M., Gasieniec, L., Rytter, W.: Fast broadcasting and gossiping in radio networks. In: FOCS 2009, pp. 575–584 (2000)Google Scholar
9. 9.
Clementi, A.E.F., Monti, A., Silvestri, R.: Distributed broadcast in radio networks of unknown topology. Theor. Comput. Sci. 302(1–3), 337–364 (2003)
10. 10.
Clifford, R., Efremenko, K., Porat, E., Rothschild, A.: Pattern matching with don’t cares and few errors. J. Comput. Syst. Sci. 76(2), 115–124 (2010)
11. 11.
Cormode, G., Muthukrishnan, S.: Combinatorial algorithms for compressed sensing. In: 40th Annual Conference on Information Sciences and Systems, pp. 198–201 (2006)Google Scholar
12. 12.
Cormode, G., Muthukrishnan, S.: What’s hot and what’s not: tracking most frequent items dynamically. ACM Trans. Database Syst. 30(1), 249–278 (2005)
13. 13.
Damaschke, P.: Threshold group testing. Electron. Notes Discrete Math. 21, 265–271 (2005)
14. 14.
DasGupta, A.: Fundamentals of Probability: A First Course. Springer Texts in Statistics. Springer, New York (2010). doi:
15. 15.
De Bonis, A., Gasieniec, L., Vaccaro, U.: Optimal two-stage algorithms for group testing problems. SIAM J. Comput. 34(5), 1253–1270 (2005)
16. 16.
De Marco, G.: Distributed broadcast in unknown radio networks. In: SODA 2008, pp. 208–217 (2008)Google Scholar
17. 17.
De Marco, G.: Distributed broadcast in unknown radio networks. SIAM J. Comput. 39(6), 2162–2175 (2010)
18. 18.
De Marco, G., Kowalski, D.R.: Contention resolution in a non-synchronized multiple access channel. Theor. Comput. Sci. (2017). https://doi.org/10.1016/j.tcs.2017.05.014
19. 19.
De Marco, G., Kowalski, D.R.: Fast nonadaptive deterministic algorithm for conflict resolution in a dynamic multiple-access channel. SIAM J. Comput. 44(3), 868–888 (2015)
20. 20.
De Marco, G., Kowalski, D.R.: Contention resolution in a non-synchronized multiple access channel. In: IPDPS 2013, pp. 525–533 (2013)Google Scholar
21. 21.
De Marco, G., Kowalski, D.R.: Towards power-sensitive communication on a multiple-access channel. In: 30th International Conference on Distributed Computing Systems (ICDCS 2010), Genoa, Italy, May 2010Google Scholar
22. 22.
De Marco, G., Pellegrini, M., Sburlati, G.: Faster deterministic wakeup in multiple access channels. Discrete Appl. Math. 155(8), 898–903 (2007)
23. 23.
De Marco, G., Kowalski, D.R.: Searching for a subset of counterfeit coins: randomization vs determinism and adaptiveness vs non-adaptiveness. Random Struct. Algorithms 42(1), 97–109 (2013)
24. 24.
Dorfman, R.: The detection of defective members of large populations. Ann. Math. Stat. 14(4), 436–440 (1943)
25. 25.
Dyachkov, A.G., Rykov, V.V.: Bounds on the length of disjunctive codes. Probl. Pereda. Informatsii 18(3), 7–13 (1982)
26. 26.
Farach, M., Kannan, S., Knill, E., Muthukrishnan, S.: Group testing problems with sequences in experimental molecular biology. In: Proceedings of the Compression and Complexity of Sequences, SEQUENCES 1997, p. 357 (1997)Google Scholar
27. 27.
Fu, H.-L., Chang, H., Shih, C.-H.: Threshold group testing on inhibitor model. J. Comput. Biol. 20(6), 464–470 (2013)
28. 28.
Indyk, P.: Deterministic superimposed coding with applications to pattern matching. In: FOCS 1997, pp. 127–136 (1997)Google Scholar
29. 29.
Indyk, P., Ngo, H.Q., Rudra, A.: Efficiently decodable non-adaptive group testing. In: SODA, pp. 1126–1142 (2010)Google Scholar
30. 30.
Kautz, W., Singleton, R.: Nonrandom binary superimposed codes. IEEE Trans. Inf. Theory 10(4), 363–377 (1964)
31. 31.
Ngo, H.Q., Du, D.-Z.: A survey on combinatorial group testing algorithms with applications to DNA library screening. In: Discrete Mathematical Problems with Medical Applications. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 55, pp. 171–182. American Mathematical Society (2000)Google Scholar
32. 32.
Ngo, H.Q., Porat, E., Rudra, A.: Efficiently decodable error-correcting list disjunct matrices and applications. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 557–568. Springer, Heidelberg (2011). doi:
33. 33.
Porat, E., Rothschild, A.: Explicit nonadaptive combinatorial group testing schemes. IEEE Trans. Inf. Theory 57(12), 7982–7989 (2011)
34. 34.
Wolf, J.: Born again group testing: multiaccess communications. IEEE Trans. Inf. Theory 31(2), 185–191 (1985)

© Springer-Verlag GmbH Germany 2017

## Authors and Affiliations

• Gianluca De Marco
• 1
Email author
• Tomasz Jurdziński
• 2
• Michał Różański
• 2
• Grzegorz Stachowiak
• 2
1. 1.Dipartimento di InformaticaUniversity of SalernoFiscianoItaly
2. 2.Institute of Computer ScienceUniversity of WrocławWrocławPoland