Efficiently Decodable Non-Adaptive Threshold Group Testing

We consider non-adaptive threshold group testing for identification of up to <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> defective items in a set of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> items, where a test is positive if it contains at least <inline-formula> <tex-math notation="LaTeX">$2 \leq u \leq d$ </tex-math></inline-formula> defective items, and negative otherwise. The defective items can be identified using <inline-formula> <tex-math notation="LaTeX">$t = O \left ({\left ({\frac {d}{u} }\right)^{u} \left ({\frac {d}{d - u} }\right)^{d-u} \left ({u \log {\frac {d}{u}} + \log {\frac {1}{\epsilon }} }\right) \cdot d^{2} \log {n} }\right)$ </tex-math></inline-formula> tests with probability at least <inline-formula> <tex-math notation="LaTeX">$1 - \epsilon $ </tex-math></inline-formula> for any <inline-formula> <tex-math notation="LaTeX">$\epsilon > 0$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$t = O \left ({\left ({\frac {d}{u} }\right)^{u} \left ({\frac {d}{d -u} }\right)^{d - u} d^{3} \log {n} \cdot \log {\frac {n}{d}} }\right)$ </tex-math></inline-formula> tests with probability 1. The decoding time is <inline-formula> <tex-math notation="LaTeX">$t \times (d^{2} \log {n})$ </tex-math></inline-formula>. This result significantly improves the best known results for decoding non-adaptive threshold group testing: <inline-formula> <tex-math notation="LaTeX">$O\left({n\log {n} + n \log {\frac {1}{\epsilon }}}\right)$ </tex-math></inline-formula> for probabilistic decoding, where <inline-formula> <tex-math notation="LaTeX">$\epsilon > 0$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$O(n^{u} \log {n})$ </tex-math></inline-formula> for deterministic decoding.

[1]  Atri Rudra,et al.  Efficiently Decodable Error-Correcting List Disjunct Matrices and Applications - (Extended Abstract) , 2011, ICALP.

[2]  D. Du,et al.  Combinatorial Group Testing and Its Applications , 1993 .

[3]  Graham Cormode,et al.  What's hot and what's not: tracking most frequent items dynamically , 2003, TODS.

[4]  Mayank Bakshi,et al.  GROTESQUE: Noisy Group Testing (Quick and Efficient) , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[5]  R. Dorfman The Detection of Defective Members of Large Populations , 1943 .

[6]  Annalisa De Bonis,et al.  An Almost Optimal Algorithm for Generalized Threshold Group Testing with Inhibitors , 2011, J. Comput. Biol..

[7]  Hung-Lin Fu,et al.  Reconstruction of hidden graphs and threshold group testing , 2011, J. Comb. Optim..

[8]  Vladimir S. Lebedev,et al.  Superimposed Codes and Threshold Group Testing , 2014, Information Theory, Combinatorics, and Search Theory.

[9]  Ely Porat,et al.  Search Methodologies , 2022 .

[10]  Mahdi Cheraghchi Improved Constructions for Non-adaptive Threshold Group Testing , 2010, ICALP.

[11]  Jack K. Wolf,et al.  Born again group testing: Multiaccess communications , 1985, IEEE Trans. Inf. Theory.

[12]  Sampath Kannan,et al.  Group testing problems with sequences in experimental molecular biology , 1997, Proceedings. Compression and Complexity of SEQUENCES 1997 (Cat. No.97TB100171).

[13]  Peter Damaschke Threshold Group Testing , 2005, Electron. Notes Discret. Math..

[14]  Mahdi Cheraghchi Noise-resilient group testing: Limitations and constructions , 2013, Discret. Appl. Math..

[15]  Hung-Lin Fu,et al.  Nonadaptive algorithms for threshold group testing , 2009, Discret. Appl. Math..

[16]  Mayank Bakshi,et al.  Stochastic threshold group testing , 2013, 2013 IEEE Information Theory Workshop (ITW).

[17]  Arkadii G. D'yachkov,et al.  Families of Finite Sets in which No Intersection of Sets Is Covered by the Union of s Others , 2002, J. Comb. Theory, Ser. A.