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Efficiently Decodable Error-Correcting List Disjunct Matrices and Applications

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Book cover Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6755))

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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 e 0 false positive and e 1 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(dlog(n/d) + e 0 + de 1) 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 error-tolerant 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(d 2logn + 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(dlogn), 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.

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References

  1. Alon, N., Asodi, V.: Tracing many users with almost no rate penalty. IEEE Trans. Inform. Theory 53(1), 437–439 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Hod, R.: Optimal monotone encodings. IEEE Transactions on Information Theory 55(3), 1343–1353 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheraghchi, M.: Noise-resilient group testing: Limitations and constructions. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds.) FCT 2009. LNCS, vol. 5699, pp. 62–73. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  4. 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)

    Article  Google Scholar 

  5. De Bonis, A., Gąsieniec, L., Vaccaro, U.: Optimal two-stage algorithms for group testing problems. SIAM J. Comput. 34(5), 1253–1270 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, D.Z., Hwang, F.K.: Combinatorial group testing and its applications, 2nd edn. Series on Applied Mathematics, vol. 12. World Scientific Publishing Co. Inc., River Edge (2000)

    MATH  Google Scholar 

  7. D’yachkov, A.G., Rykov, V.V.: A survey of superimposed code theory. Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 12(4), 229–242 (1983)

    MathSciNet  MATH  Google Scholar 

  8. D’yachkov, A.G., Rykov, V.V., Rashad, A.M.: Superimposed distance codes. Problems Control Inform. Theory 18(4), 237–250 (1989)

    MathSciNet  MATH  Google Scholar 

  9. 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(1-2), 79–89 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Füredi, Z.: On r-cover-free families. J. Combin. Theory Ser. A 73(1), 172–173 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ganguly, S.: Data stream algorithms via expander graphs. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 52–63. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Goodrich, M.T., Atallah, M.J., Tamassia, R.: Indexing information for data forensics. In: Third International Conference on Applied Cryptography and Network Security (ANCS), pp. 206–221 (2005)

    Google Scholar 

  13. Indyk, P.: Explicit constructions of selectors and related combinatorial structures, with applications. In: SODA, pp. 697–704 (2002)

    Google Scholar 

  14. Indyk, P., Ngo, H.Q., Rudra, A.: Efficiently decodable non-adaptive group testing. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1126–1142 (2010)

    Google Scholar 

  15. Kainkaryam: Pooling in high-throughput drug screening. Current Opinion in Drug Discovery & Development 12(3), 339–350 (2009)

    Google Scholar 

  16. Khattab, S.M., Gobriel, S., Melhem, R.G., Mossé, D.: Live baiting for service-level dos attackers. In: INFOCOM, pp. 171–175 (2008)

    Google Scholar 

  17. Moran, T., Naor, M., Segev, G.: Deterministic history-independent strategies for storing information on write-once memories. Theory of Computing 5(1), 43–67 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 Ser. Discrete Math. Theoret. Comput. Sci., vol. 55, pp. 171–182. Amer. Math. Soc., Providence (2000)

    Google Scholar 

  19. Parvaresh, F., Vardy, A.: Correcting errors beyond the Guruswami-Sudan radius in polynomial time. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 285–294 (2005)

    Google Scholar 

  20. Porat, E., Rothschild, A.: Explicit non-adaptive combinatorial group testing schemes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 748–759. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  21. Rashad, A.M.: Random coding bounds on the rate for list-decoding superimposed codes. Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 19(2), 141–149 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Rudra, A., Uurtamo, S.: Data stream algorithms for codeword testing. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 629–640. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of CSE, University at Buffalo, SUNY, Buffalo, NY, 14260, USA

    Hung Q. Ngo & Atri Rudra

  2. Department of Computer Science, Bar-Ilan University, Ramat Gan, 52900, Israel

    Ely Porat

Authors
  1. Hung Q. Ngo
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  2. Ely Porat
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  3. Atri Rudra
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Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

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Ngo, H.Q., Porat, E., Rudra, A. (2011). Efficiently Decodable Error-Correcting List Disjunct Matrices and Applications. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_47

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  • DOI: https://doi.org/10.1007/978-3-642-22006-7_47

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22005-0

  • Online ISBN: 978-3-642-22006-7

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