Efficiently Decodable Error-Correcting List Disjunct Matrices and Applications - (Extended Abstract)

A (d, l)-list disjunct matrix is a non-adaptive group testing primitive which, given a set of items with at most d "defectives," outputs a superset of the defectives containing less than l non-defective items. The primitive has found many applications as stand alone objects and as building blocks in the construction of other combinatorial objects. This paper studies error-tolerant list disjunct matrices which can correct up to e0 false positive and e1 false negative tests in sub-linear time. We then use list-disjunct matrices to prove new results in three different applications. Our major contributions are as follows. (1) We prove several (almost)- matching lower and upper bounds for the optimal number of tests, including the fact that Θ(d log(n/d) + e0+ de1) tests is necessary and sufficient when l = Θ(d). Similar results are also derived for the disjunct matrix case (i.e. l = 1). (2) We present two methods that convert errortolerant list disjunct matrices in a black-box manner into error-tolerant list disjunct matrices that are also efficiently decodable. The methods help us derive a family of (strongly) explicit constructions of list-disjunct matrices which are either optimal or near optimal, and which are also efficiently decodable. (3) We show how to use error-correcting efficiently decodable list-disjunct matrices in three different applications: (i) explicit constructions of d-disjunct matrices with t = O(d2 log n+rd) tests which are decodable in poly(t) time, where r is the maximum number of test errors. This result is optimal for r = Ω(d log n), and even for r = 0 this result improves upon known results; (ii) (explicit) constructions of (near)- optimal, error-correcting, and efficiently decodable monotone encodings; and (iii) (explicit) constructions of (near)-optimal, error-correcting, and efficiently decodable multiple user tracing families.

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

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

[3]  Mikhail J. Atallah,et al.  Indexing Information for Data Forensics , 2005, ACNS.

[4]  Arkadii G. D'yachkov,et al.  A survey of superimposed code theory , 1983 .

[5]  Atri Rudra,et al.  Data Stream Algorithms for Codeword Testing , 2010, ICALP.

[6]  Annalisa De Bonis,et al.  Optimal Two-Stage Algorithms for Group Testing Problems , 2005, SIAM J. Comput..

[7]  Noga Alon,et al.  Optimal Monotone Encodings , 2009, IEEE Transactions on Information Theory.

[8]  Noga Alon,et al.  Tracing Many Users With Almost No Rate Penalty , 2007, IEEE Transactions on Information Theory.

[9]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[10]  Robin Milner.  An Action Structure for Synchronous pi-Calculus , 1993, FCT.

[11]  Sumit Ganguly,et al.  Data Stream Algorithms via Expander Graphs , 2008, ISAAC.

[12]  Panos M. Pardalos,et al.  Discrete Mathematical Problems with Medical Applications , 2000 .

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

[14]  Rami G. Melhem,et al.  Live Baiting for Service-Level DoS Attackers , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.

[15]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[16]  P. Erdös,et al.  Families of finite sets in which no set is covered by the union ofr others , 1985 .

[17]  Miklós Ruszinkó,et al.  On the Upper Bound of the Size of the R-Cover-Free Families , 1993, Proceedings. IEEE International Symposium on Information Theory.

[18]  Piotr Indyk,et al.  Explicit constructions of selectors and related combinatorial structures, with applications , 2002, SODA '02.

[19]  Atri Rudra,et al.  Efficiently decodable non-adaptive group testing , 2010, SODA '10.

[20]  Raghunandan M Kainkaryam,et al.  Pooling in high-throughput drug screening. , 2009, Current opinion in drug discovery & development.

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

[22]  Zoltán Füredi.  On r-Cover-free Families , 1996, J. Comb. Theory, Ser. A.

[23]  Ding-Zhu Du,et al.  A survey on combinatorial group testing algorithms with applications to DNA Library Screening , 1999, Discrete Mathematical Problems with Medical Applications.

[24]  V. V. Rykov,et al.  Superimposed distance codes , 1989 .

[25]  Moni Naor,et al.  Deterministic History-Independent Strategies for Storing Information on Write-Once Memories , 2007, Theory Comput..