New combinatorial structures with applications to efficient group testing with inhibitors
Group testing with inhibitors (GTI) is a variant of classical group testing where in addition to positive items and negative items, there is a third class of items called inhibitors . In this model the response to a test is YES if and only if the tested group of items contains at least one positive...
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| Veröffentlicht in: | Journal of combinatorial optimization Jg. 15; H. 1; S. 77 - 94 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.01.2008
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| ISSN: | 1382-6905, 1573-2886 |
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| Abstract | Group testing with inhibitors
(GTI) is a variant of classical group testing where in addition to positive items and negative items, there is a third class of items called
inhibitors
. In this model the response to a test is YES if and only if the tested group of items contains at least one positive item and no inhibitor. This model of group testing has been introduced by Farach et al. (Proceedings of compression and complexity of sequences, pp 357–367,
1997
) for applications in the field of molecular biology. In this paper we investigate the GTI problem both in the case when the exact number of positive items is given, and in the case when the number of positives is not given but we are provided with an upper bound on it. For the latter case, we present a lower bound on the number of tests required to determine the positive items in a completely nonadaptive fashion. Also under the same hypothesis, we derive an improved lower bound on the number of tests required by
any
algorithm (using any number of stages) for the GTI problem.
As far as it concerns the case when the exact number of positives is known, we give an efficient trivial two-stage algorithm. Instrumental to our results are new combinatorial structures introduced in this paper. In particular we introduce generalized versions of the well known superimposed codes (Du, D.Z., Hwang, F.K. in Pooling designs and nonadaptive group testing,
2006
; Dyachkov, A.G., Rykov, V.V. in Probl. Control Inf. Theory 12:7–13,
1983
; Dyachkov, A.G., et al. in J. Comb. Theory Ser. A 99:195–218,
2002
; Kautz, W.H., Singleton, R.R. in IEEE Trans. Inf. Theory 10:363–377,
1964
) and selectors (Clementi, A.E.F, et al. in Proceedings of symposium on discrete algorithms, pp. 709–718,
2001
; De Bonis, A., et al. in SIAM J Comput. 34(5):1253–1270,
2005
; Indyk, P. in Proceedings of symposium on discrete algorithms, pp. 697–704,
2002
) that we believe to be of independent interest. |
|---|---|
| AbstractList | Group testing with inhibitors
(GTI) is a variant of classical group testing where in addition to positive items and negative items, there is a third class of items called
inhibitors
. In this model the response to a test is YES if and only if the tested group of items contains at least one positive item and no inhibitor. This model of group testing has been introduced by Farach et al. (Proceedings of compression and complexity of sequences, pp 357–367,
1997
) for applications in the field of molecular biology. In this paper we investigate the GTI problem both in the case when the exact number of positive items is given, and in the case when the number of positives is not given but we are provided with an upper bound on it. For the latter case, we present a lower bound on the number of tests required to determine the positive items in a completely nonadaptive fashion. Also under the same hypothesis, we derive an improved lower bound on the number of tests required by
any
algorithm (using any number of stages) for the GTI problem.
As far as it concerns the case when the exact number of positives is known, we give an efficient trivial two-stage algorithm. Instrumental to our results are new combinatorial structures introduced in this paper. In particular we introduce generalized versions of the well known superimposed codes (Du, D.Z., Hwang, F.K. in Pooling designs and nonadaptive group testing,
2006
; Dyachkov, A.G., Rykov, V.V. in Probl. Control Inf. Theory 12:7–13,
1983
; Dyachkov, A.G., et al. in J. Comb. Theory Ser. A 99:195–218,
2002
; Kautz, W.H., Singleton, R.R. in IEEE Trans. Inf. Theory 10:363–377,
1964
) and selectors (Clementi, A.E.F, et al. in Proceedings of symposium on discrete algorithms, pp. 709–718,
2001
; De Bonis, A., et al. in SIAM J Comput. 34(5):1253–1270,
2005
; Indyk, P. in Proceedings of symposium on discrete algorithms, pp. 697–704,
2002
) that we believe to be of independent interest. |
| Author | De Bonis, Annalisa |
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| Keywords | Group testing algorithms Superimposed codes Computational molecular biology Selectors Pooling designs |
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| References | Chrobak M, Gasieniec L, Rytter W (2000) Fast broadcasting and gossiping in radio networks. In: Proceedings of 42nd IEEE annual symposium on foundation of computer science (FOCS 2000), pp 575–581 LiCHA sequential method for screening experimental variablesJ Am Stat Assoc1962574554770105.1220210.2307/2281652 BergerTMehravariNTowsleyDWolfJRandom multiple-access communication and group testingIEEE Trans Commun198432776977910.1109/TCOM.1984.1096146 Dyachkov AG, Macula AJ, Torney DC, Vilenkin PA (2001) Two models of nonadaptive group testing for designing screening experiments. In: Proceedings of the 6th international workshop on model-oriented designs and analysis, pp 63–75 Indyk P (2002) Explicit constructions of selectors and related combinatorial structures, with applications. In: Proceedings of symposium on discrete algorithms 2002 (SODA 2002), pp 697–704 LovàszLOn the ratio of optimal integral and fractional coversDiscret Math1975133833900323.0512710.1016/0012-365X(75)90058-8 Indyk P (1997) Deterministic superimposed coding with application to pattern matching. In: Proceedings of thirty-ninth IEEE annual symposium on foundations of computer science (FOCS 97), pp 127–136 FarachMKannanSKnillEHMuthukrishnanSCarpentieriBDe SantisAVaccaroUStorerJGroup testing with sequences in experimental molecular biologyProceedings of compression and complexity of sequences1997Los AlamitosIEEE Computer Society357367 BaldingDJBrunoWJKnillETorneyDCSpeedTPWatermanMSA comparative survey of non-adaptive pooling designGenetic mapping and DNA sequencing1996BerlinSpringer133154 StinsonDRvan TrungTWeiRSecure frameproof codes, key distribution patterns, group testing algorithms and related structuresJ Stat Plan Inference2000865956171054.9401310.1016/S0378-3758(99)00131-7 DuDZHwangFKPooling designs and nonadaptive group testing2006SingaporeWorld Scientific05070274 NgoHQDuD-ZA survey on combinatorial group testing algorithms with applications to DNA library screeningDiscrete mathematical problems with medical applications2000ProvidenceAmerican Mathematical Society171182 DyachkovAGRykovVVA survey of superimposed code theoryProbl Control Inf Theory1983124113729278 KautzWHSingletonRRNonrandom binary superimposed codesIEEE Trans Inf Theory1964103633770133.1240210.1109/TIT.1964.1053689 DyachkovAGVilenkinPMaculaATorneyDFamilies of finite sets in which no intersections of ℓ sets is covered by the union of s othersJ Comb Theory Ser A20029919521810.1006/jcta.2002.32571917018 SobelMGrollPAGroup testing to eliminate efficiently all defectives in a binomial sampleBell Syst Tech J19593811791252107341 De BonisAGasieniecLVaccaroUOptimal two-stage algorithms for group testing problemsSIAM J Comput2005345125312701079.6804310.1137/S00975397034280022165946 Knill E (1995) Lower bounds for identifying subset members with subset queries. In: Proceedings of symposium on discrete algorithms 1995 (SODA 1995), pp 369–377 WolfJBorn again group testing: multiaccess communicationsIEEE Trans Inf Theory1985311851910586.9401110.1109/TIT.1985.1057026 DuDZHwangFKCombinatorial group testing and its applications2000SingaporeWorld Scientific0952.90001 Chaudhuri S, Radhakrishnan J (1996) Deterministic restrictions in circuit complexity. In: Proceedings of the twenty-eighth annual ACM symposium on the theory of computing (STOC 96), pp 30–36 Barillot E, Lacroix B, Cohen D (1991) Theoretical analysis of library screening using an n-dimensional pooling strategy. In: Nucleic acids research, pp 6241–6247 Clementi AEF, Monti A, Silvestri R (2001) Selective families, superimposed codes, and broadcasting on unknown radio networks. In: Proceedings of symposium on discrete algorithms (SODA’01), pp 709–718 HwangFKLiuYCError-tolerant pooling designs with inhibitorsJ Comput Biol200310223123610.1089/106652703321825982 PevznerPALipshutzRTowards DNA sequencing chips19th international conference on mathematical foundations of computer science1994BerlinSpringer143158 Hong YW, Scaglione A (2004) On multiple access for distributed dependent sensors: a content-based group testing approach. In: IEEE information theory workshop, pp 298–303 StinsonDRWeiRZhuLSome new bounds for cover-free familiesJ Comb Theory Ser A2000902242340948.0505510.1006/jcta.1999.30361749434 Margaritis D, Skiena S (1995) Reconstructing strings from substrings in rounds. In: Proceedings of thirty-seventh IEEE annual symposium on foundations of computer science (FOCS 95), pp 613–620 ErdösPFranklPFürediZFamilies of finite sets in which no set is covered by the union of r othersIsrael J Math198551758910.1007/BF02772959 HongEHLadnerREGroup testing for image compressionIEEE Trans Image Process20021190191110.1109/TIP.2002.801124 De BonisAVaccaroUImproved algorithms for group testing with inhibitorsInf Process Lett199866576410.1016/S0020-0190(98)00088-X DorfmanRThe detection of defective members of large populationsAnn Math Stat194314436440 KumarRRajagopalanSSahaiACoding constructions for blacklisting problems without computational assumptionsProceedings of CRYPTO ‘991999BerlinSpringer609623 BrunoWJBaldingDJKnillEBruceDWhittakerCDoggetNStallingRTorneyDCDesign of efficient pooling experimentsGenomics199526213010.1016/0888-7543(95)80078-Z 9085_CR14 9085_CR1 M Farach (9085_CR17) 1997 9085_CR6 9085_CR7 PA Pevzner (9085_CR30) 1994 9085_CR5 DR Stinson (9085_CR33) 2000; 86 DJ Balding (9085_CR2) 1996 DZ Du (9085_CR12) 2006 AG Dyachkov (9085_CR13) 1983; 12 J Wolf (9085_CR34) 1985; 31 R Dorfman (9085_CR10) 1943; 14 DZ Du (9085_CR11) 2000 EH Hong (9085_CR18) 2002; 11 R Kumar (9085_CR25) 1999 L Lovàsz (9085_CR27) 1975; 13 AG Dyachkov (9085_CR15) 2002; 99 9085_CR28 WH Kautz (9085_CR23) 1964; 10 9085_CR24 M Sobel (9085_CR31) 1959; 38 9085_CR21 FK Hwang (9085_CR20) 2003; 10 9085_CR22 T Berger (9085_CR3) 1984; 32 P Erdös (9085_CR16) 1985; 51 WJ Bruno (9085_CR4) 1995; 26 DR Stinson (9085_CR32) 2000; 90 A Bonis De (9085_CR9) 2005; 34 HQ Ngo (9085_CR29) 2000 CH Li (9085_CR26) 1962; 57 A Bonis De (9085_CR8) 1998; 66 9085_CR19 |
| References_xml | – reference: DyachkovAGRykovVVA survey of superimposed code theoryProbl Control Inf Theory1983124113729278 – reference: Margaritis D, Skiena S (1995) Reconstructing strings from substrings in rounds. In: Proceedings of thirty-seventh IEEE annual symposium on foundations of computer science (FOCS 95), pp 613–620 – reference: LovàszLOn the ratio of optimal integral and fractional coversDiscret Math1975133833900323.0512710.1016/0012-365X(75)90058-8 – reference: BergerTMehravariNTowsleyDWolfJRandom multiple-access communication and group testingIEEE Trans Commun198432776977910.1109/TCOM.1984.1096146 – reference: Chrobak M, Gasieniec L, Rytter W (2000) Fast broadcasting and gossiping in radio networks. In: Proceedings of 42nd IEEE annual symposium on foundation of computer science (FOCS 2000), pp 575–581 – reference: Clementi AEF, Monti A, Silvestri R (2001) Selective families, superimposed codes, and broadcasting on unknown radio networks. In: Proceedings of symposium on discrete algorithms (SODA’01), pp 709–718 – reference: DuDZHwangFKCombinatorial group testing and its applications2000SingaporeWorld Scientific0952.90001 – reference: Hong YW, Scaglione A (2004) On multiple access for distributed dependent sensors: a content-based group testing approach. In: IEEE information theory workshop, pp 298–303 – reference: KautzWHSingletonRRNonrandom binary superimposed codesIEEE Trans Inf Theory1964103633770133.1240210.1109/TIT.1964.1053689 – reference: ErdösPFranklPFürediZFamilies of finite sets in which no set is covered by the union of r othersIsrael J Math198551758910.1007/BF02772959 – reference: HwangFKLiuYCError-tolerant pooling designs with inhibitorsJ Comput Biol200310223123610.1089/106652703321825982 – reference: BaldingDJBrunoWJKnillETorneyDCSpeedTPWatermanMSA comparative survey of non-adaptive pooling designGenetic mapping and DNA sequencing1996BerlinSpringer133154 – reference: FarachMKannanSKnillEHMuthukrishnanSCarpentieriBDe SantisAVaccaroUStorerJGroup testing with sequences in experimental molecular biologyProceedings of compression and complexity of sequences1997Los AlamitosIEEE Computer Society357367 – reference: BrunoWJBaldingDJKnillEBruceDWhittakerCDoggetNStallingRTorneyDCDesign of efficient pooling experimentsGenomics199526213010.1016/0888-7543(95)80078-Z – reference: Indyk P (1997) Deterministic superimposed coding with application to pattern matching. In: Proceedings of thirty-ninth IEEE annual symposium on foundations of computer science (FOCS 97), pp 127–136 – reference: Barillot E, Lacroix B, Cohen D (1991) Theoretical analysis of library screening using an n-dimensional pooling strategy. In: Nucleic acids research, pp 6241–6247 – reference: De BonisAGasieniecLVaccaroUOptimal two-stage algorithms for group testing problemsSIAM J Comput2005345125312701079.6804310.1137/S00975397034280022165946 – reference: De BonisAVaccaroUImproved algorithms for group testing with inhibitorsInf Process Lett199866576410.1016/S0020-0190(98)00088-X – reference: HongEHLadnerREGroup testing for image compressionIEEE Trans Image Process20021190191110.1109/TIP.2002.801124 – reference: StinsonDRWeiRZhuLSome new bounds for cover-free familiesJ Comb Theory Ser A2000902242340948.0505510.1006/jcta.1999.30361749434 – reference: StinsonDRvan TrungTWeiRSecure frameproof codes, key distribution patterns, group testing algorithms and related structuresJ Stat Plan Inference2000865956171054.9401310.1016/S0378-3758(99)00131-7 – reference: Chaudhuri S, Radhakrishnan J (1996) Deterministic restrictions in circuit complexity. In: Proceedings of the twenty-eighth annual ACM symposium on the theory of computing (STOC 96), pp 30–36 – reference: Dyachkov AG, Macula AJ, Torney DC, Vilenkin PA (2001) Two models of nonadaptive group testing for designing screening experiments. In: Proceedings of the 6th international workshop on model-oriented designs and analysis, pp 63–75 – reference: DyachkovAGVilenkinPMaculaATorneyDFamilies of finite sets in which no intersections of ℓ sets is covered by the union of s othersJ Comb Theory Ser A20029919521810.1006/jcta.2002.32571917018 – reference: KumarRRajagopalanSSahaiACoding constructions for blacklisting problems without computational assumptionsProceedings of CRYPTO ‘991999BerlinSpringer609623 – reference: Indyk P (2002) Explicit constructions of selectors and related combinatorial structures, with applications. In: Proceedings of symposium on discrete algorithms 2002 (SODA 2002), pp 697–704 – reference: Knill E (1995) Lower bounds for identifying subset members with subset queries. 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(GTI) is a variant of classical group testing where in addition to positive items and negative items, there is a third class of... |
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| SubjectTerms | Combinatorics Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation |
| Title | New combinatorial structures with applications to efficient group testing with inhibitors |
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