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Tracing a single user. (English) Zbl 1105.05068
Denote \(g(n,r)\) the maximum possible cardinality of a family of subsets of an \(n\)-element underlying set so that given a union of at most \(r\) members, one can identify at least one of the members. This has a close connection with the so-called group testing problem applied to design DNA chips in molecular biology. The paper shows that \(g(n,r)=2^{\Theta({n \over r})}.\) The main contribution of the paper is a new lower bound for \(g(n,r)\).

05D05 Extremal set theory
92D20 Protein sequences, DNA sequences
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[1] Alon, N.; Babai, L.; Itai, A., A fast and simple randomized parallel algorithm for the maximal independent set problem, J. algorithms, 7, 567-583, (1986) · Zbl 0631.68063
[2] N. Alon, D. Moshkovitz, M. Safra, Algorithmic construction of sets for \(k\)-restrictions, ACM Trans. Algorithms (in press) · Zbl 1321.68445
[3] Alon, N.; Beigel, R.; Kasif, S.; Rudich, S.; Sudakov, B., Learning a hidden matching, (), SIAM J. computing, 33, 487-501, (2004), Also: · Zbl 1055.05142
[4] R. Beigel, N. Alon, M.S. Apaydin, L. Fortnow, S. Kasif, An optimal procedure for gap closing in whole genome shotgun sequencing, in: Proc. 2001 RECOMB, ACM Press, pp. 22-30
[5] Balding, D.J.; Bruno, W.J.; Knill, E.; Torney, D.C., A comparative survey of non-adaptive pooling designs, genetic mapping and DNA sequencing, (), 133-154 · Zbl 0860.62083
[6] Colbourn, C.C.; Ling, A.C.H.; Tompa, M., Construction of optimal quality control for oligo arrays, Bioinformatics, 18, 4, 529-535, (2002)
[7] Csűrös, M.; Ruszinkó, M., Single-user tracing and disjointly superimposed codes, IEEE trans. inform. theory, 51, 4, 1606-1611, (2005) · Zbl 1284.94026
[8] Coppersmith, D.; Shearer, J.B., New bounds for union-free families of sets, Electron. J. combin., 5, 1, #R39, (1998)
[9] Dyachkov, A.G.; Rykov, V.V., Bounds on the length of disjunctive codes, Probl. pereda. inf., 18, 3, 158-166, (1982) · Zbl 0524.94015
[10] Erdős, P.; Frankl, P.; Füredi, Z., Families of finite sets in which no set is covered by the union of r others, Israel J. math., 51, 79-89, (1985) · Zbl 0587.05021
[11] Füredi, Z., A note on r-cover-free families, J. combin. theory ser. A, 73, 1, 172-173, (1996) · Zbl 0843.05100
[12] Hwang, F.K.; Sós, V.T., Non-adaptive hypergeometric group testing, Studia sci. math. hungar., 22, 257-263, (1987) · Zbl 0639.62076
[13] Kautz, W.H.; Singleton, R.C., Nonrandom binary superimposed codes, IEEE trans. inform. theory, 10, 363-377, (1964) · Zbl 0133.12402
[14] Ruszinkó, M., On the upper bound of the size of the r-cover-free families, J. combin. theory ser. A, 66, 2, 302-310, (1994) · Zbl 0798.05071
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