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Linear Time Constructions of Some \(d\)-Restriction Problems

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Algorithms and Complexity (CIAC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9079))

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Abstract

We give new linear time globally explicit constructions for perfect hash families, cover-free families and separating hash functions.

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References

  1. Alon, N.: Explicit construction of exponential sized families of k-independent sets. Discrete Math. 58, 191–193 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alon, N., Asodi, V.: Learning a hidden subgraph. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 110–121. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. Alon, N., Beigel, R., Kasif, S., Rudich, S., Sudakov, B.: Learning a Hidden Matching. SIAM J. Comput. 33(2), 487–501 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Angluin, D., Chen, J.: Learning a Hidden Hypergraph. Journal of Machine Learning Research 7, 2215–2236 (2006)

    MATH  MathSciNet  Google Scholar 

  5. Angluin, D., Chen, J.: Learning a hidden graph using \(O(\log n)\) queries per edge. J. Comput. Syst. Sci. 74(4), 546–556 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alon, N., Naor, M.: Rerandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions. Algorithmica 16, 434–449 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Alon, N., Moshkovitz, D., Safra, S.: Algorithmic construction of sets for \(k\)-restrictions. ACM Transactions on Algorithms 2(2), 153–177 (2006)

    Article  MathSciNet  Google Scholar 

  8. Atici, M., Magliveras, S.S., Stinson, D.R., Wei, W.-D.: Some Recursive Constructions for Perfect Hash Families. Journal of Combinatorial Designs 4(5), 353–363 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Alon, N., Bruck, J., Naor, J., Naor, M., Roth, R.M.: Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory 38(2), 509–516 (1992)

    Article  MATH  Google Scholar 

  10. Blackburn, S.R.: Perfect Hash Families: Probabilistic Methods and Explicit Constructions. Journal of Combinatorial Theory, Series A 92, 54–60 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Blackburn, S.R.: Combinatorics and threshold cryptography. In: Holroyd, F.C., Quinn, K.A.S., Rowley, C., Webb, B.S. (eds.) Combinatorial Designs and Their Applications. Research Notes in Mathematics, vol. 403, pp. 44–70. CRC Press, London (1999)

    Google Scholar 

  12. Blackburn, S.R., Burmester, M., Desmedt, Y.G., Wild, P.R.: Efficient multiplicative sharing schemes. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 107–118. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  13. Bshouty, N.H.: Testers and their applications. In: ITCS 2014, pp. 327–352 (2014). Full version: Electronic Colloquium on Computational Complexity (ECCC) 19, 11 (2012)

    Google Scholar 

  14. Bshouty, N.H.: Dense Testers: Almost Linear Time and Locally Explicit Constructions. Electronic Colloquium on Computational Complexity (ECCC) 22: 6 (2015)

    Google Scholar 

  15. Bshouty, N.H.: Linear time Constructions of some \(d\)-Restriction Problems. CoRR abs/1406.2108. (2014)

    Google Scholar 

  16. Beigel, R., Alon, N., Kasif, S., Apaydin, M.S., Fortnow, L.: An optimal procedure for gap closing in whole genome shotgun sequencing. RECOMB 2001, 22–30 (2001)

    Google Scholar 

  17. Blackburn, S.R., Etzion, T., Stinson, D.R., Zaverucha, G.M.: A bound on the size of separating hash families. Journal of Combinatorial Theory, Series A 115(7), 1246–1256 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bouvel, M., Grebinski, V., Kucherov, G.: Combinatorial Search on Graphs Motivated by Bioinformatics Applications: A Brief Survey. WG 2005, 16–27 (2005)

    MathSciNet  Google Scholar 

  19. Baker, R.C., Harman, G., Pintz, J.: The difference between consecutive primes. II. Proceedings of the London Mathematical Society 83(3), 532–562 (2001)

    Google Scholar 

  20. Barwick, S.G., Jackson, W.-A.: Geometric constructions of optimal linear perfect hash families. Finite Fields and Their Applications 14(1), 1–13 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Barwick, S.G., Jackson, W.-A., Quinn, C.T.: Optimal Linear Perfect Hash Families with Small Parameters. Journal of Combinatorial Designs 12(5), 311–324 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Bazrafshan, M., van Trung, T.: Bounds for separating hash families. Journal of Combinatorial Theory, Series A 118(3), 1129–1135 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Blackburn, S.R., Wild, P.R.: Optimal linear perfect hash families. Journal of Combinatorial Theory, Series A 83(2), 233–250 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Chang, H., Chen, H.-B., Fu, H.-L., Shi, C.-H.: Reconstruction of hidden graphs and threshold group testing. J. Comb. Optim. 22(2), 270–281 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Canetti, R., Garay, J., Itkis, G., Micciancio, D., Naor, M., Pinkas, B.: Multicast security: a taxonomy and some efficient constructions. In: Proceedings of INFOCOM 1999, vol. 2, pp. 708–716 (1999)

    Google Scholar 

  26. Chen, H.-B., Hwang, F.K.: A survey on nonadaptive group testing algorithms through the angle of decoding. J. Comb. Optim. 15(1), 49–59 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Czech, Z.J., Havas, G., Majewski, B.S.: Perfect hashing. Theoret. Comput. Sci. 182, 1–143 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Chin, F.Y.L., Leung, H.C.M., Yiu, S.-M.: Non-adaptive complex group testing with multiple positive sets. Theor. Comput. Sci. 505, 11–18 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Dyer, M., Fenner, T., Frieze, A., Thomason, A.: On key storage in secure networks. J. Cryptol. 8, 189–200 (1995)

    Article  MATH  Google Scholar 

  30. Du, D.Z., Hwang, F.K.: Combinatorial group testing and its applications. Series on Applied Mathematics. 2nd edn., vol. 12. World Scientific, New York (2000)

    Google Scholar 

  31. Dýachkov, A.G., Rykov, V.V.: Bounds on the length of disjunctive codes. Problemy Peredachi Inf. 18(3), 7–13 (1982)

    Google Scholar 

  32. Dýachkov, A.G., Rykov, V.V., Rashad, A.M.: Superimposed distance codes. Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 18(4), 237–250 (1989)

    MATH  MathSciNet  Google Scholar 

  33. Desmedt, Y., Safavi-Naini, R., Wang, H., Batten, L., Charnes, C., Pieprzyk, J.: Broadcast anti-jamming systems. Comput. Networks 35, 223–236 (2001)

    Google Scholar 

  34. D’yachkov, A., Vilenkin, P., Macula, A., Torney, D.: Families of finite sets in which no intersection of \(\ell \) sets is covered by the union of \(s\) others. J. Comb. Theory Ser. A. 99, 195–218 (2002)

    Google Scholar 

  35. Gao, H., Hwang, F.K., Thai, M.T., Wu, W., Znati, T.: Construction of d(H)-disjunct matrix for group testing in hypergraphs. J. Comb. Optim. 12(3), 297–301 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Füredi, Z.: On r-cover-free families. Journal of Combinatorial Theory, Series A 73(1), 172–173 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  37. Fredman, M.L., Komlós, J.: On the size of seperating systems and families of perfect hash function. SIAM J. Algebraic and Discrete Methods 5(1), 61–68 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  38. Fomin, F.V., Lokshtanov, D., Saurabh, S.: Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms. SODA 2014, 142–151 (2014)

    Google Scholar 

  39. Fiat, A., Naor, M.: Broadcast encryption. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 480–491. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  40. Grebinski, V., Kucherov, G.: Reconstructing a Hamiltonian Cycle by Querying the Graph: Application to DNA Physical Mapping. Discrete Applied Mathematics 88(1–3), 147–165 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  41. Garay, J.A., Staddon, J., Wool, A.: Long-Lived Broadcast Encryption. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 333–352. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  42. Huang, T., Wang, K., Weng, C.-W.: A class of error-correcting pooling designs over complexes. J. Comb. Optim. 19(4), 486–491 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  43. Indyk, P., Ngo, H.Q., Rudra, A.: Efficiently decodable non-adaptive group testing. In: The 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 1126–1142 (2010)

    Google Scholar 

  44. Körner, J.: Fredman-Komlós bounds and information theory. SIAM J. Algebraic and Discrete Methods 7(4), 560–570 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  45. Körner, J., Marton, K.: New bounds for perfect hashing via information theory. Europ. J. of Combinatorics 9(6), 523–530 (1988)

    Article  MATH  Google Scholar 

  46. Kumar, R., Rajagopalan, S., Sahai, A.: Coding constructions for blacklisting problems without computational assumptions. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 609–623. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  47. Kautz, W.H., Singleton, R.C.: Nonrandom binary superimposed codes. IEEE Trans. Inform. Theory 10(4), 363–377 (1964)

    Article  MATH  Google Scholar 

  48. Liu, L., Shen, H.: Explicit constructions of separating hash families from algebraic curves over finite fields. Designs, Codes and Cryptography 41(2), 221–233 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  49. Martirosyan, S.: Perfect Hash Families, Identifiable Parent Property Codes and Covering Arrays. Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (2003)

    Google Scholar 

  50. Mehlborn, K.: Data Structures and Algorithms. 1. Sorting and Searching. Springer, Berlin (1984)

    Google Scholar 

  51. Mehlhorn, K.: On the program size of perfect and universal hash functions. In: Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science (FOCS 1982), pp. 170–175 (1982)

    Google Scholar 

  52. Mitchell, C.J., Piper, F.C.: Key storage in secure networks. Discrete Appl. Math. 21, 215–228 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  53. Macula, A.J., Popyack, L.J.: A group testing method for finding patterns in data. Discret Appl Math. 144, 149–157 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  54. Martirosyan, S., van Trung, T.: Explicit constructions for perfect hash families. Designs, Codes and Cryptography 46(1), 97–112 (2008)

    Article  MathSciNet  Google Scholar 

  55. Nilli, A.: Perfect hashing and probability. Combinatorics, Probability and Computing 3(3), 407–409 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  56. Ngo, H.Q., Du, D.Z.: A survey on combinatorial group testing algorithms with applications to DNA library screening. Theoretical Computer Science 55, 171–182 (2000)

    MathSciNet  Google Scholar 

  57. Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  58. Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and Near-optimal Derandomization. FOCS 95, 182–191 (1995)

    Google Scholar 

  59. Newman, I., Wigderson, A.: Lower bounds on formula size of Boolean functions using hypergraph entropy. SIAM J. Discrete Math. 8, 536–542 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  60. Porat, E., Rothschild, A.: Explicit Nonadaptive Combinatorial Group Testing Schemes. IEEE Transactions on Information Theory 57(12), 7982–7989 (2011)

    Article  MathSciNet  Google Scholar 

  61. Reyzin, L., Srivastava, N.: Learning and verifying graphs using queries with a focus on edge counting. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds.) ALT 2007. LNCS (LNAI), vol. 4754, pp. 285–297. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  62. Stinson, D.R.: On some methods for unconditionally secure key distribution and broadcast encryption. Des. Codes Cryptogr. 12, 215–243 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  63. Stinson, D.R., Wei, R.: Combinatorial properties and constructions of traceability schemes and frameproof codes. SIAM J. Discrete Math. 11, 41–53 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  64. Safavi-Naini, R., Wang, H.: Multireceiver authentication codes: models, bounds, constructions, and extensions Inform. Comput. 151, 148–172 (1999)

    MATH  MathSciNet  Google Scholar 

  65. Stinson, D.R., Wei, R., Zhu, L.: New constructions for perfect hash families and related structures using combintorial designs and codes. J. Combin. Designs. 8(3), 189–200 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  66. Stinson, D.R., Wei, R., Chen, K.: On generalised separating hash families. Journal of Combinatorial Theory, Series A 115(1), 105–120 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  67. Stinson, D.R., Wei, R., Zhu, L.: Some new bounds for cover-free families. Journal of Combinatorial Theory, Series A 90(1), 224–234 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  68. Stinson, D.R., van Trung, T.: Some new results on key distribution patterns and broadcast encryption. Des. Codes Cryptogr. 14, 261–279 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  69. Stinson, D.R., van Trung, T., Wei, R.: Secure frameproof codes, key distribution patterns, group testing algorithms and related structures. J. Stat. Planning and Inference 86(2), 595–617 (2000)

    Article  MATH  Google Scholar 

  70. Torney, D.C.: Sets pooling designs. Ann. Comb. 3, 95–101 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  71. Wang, H., Xing, C.P.: Explicit Constructions of perfect hash families from algebraic curves over finite fields. J. of Combinatorial Theory, Series A 93(1), 112–124 (2001)

    Google Scholar 

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Authors and Affiliations

  1. Technion, Haifa, Israel

    Nader H. Bshouty

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  1. Nader H. Bshouty
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Correspondence to Nader H. Bshouty .

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  1. LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France

    Vangelis Th. Paschos

  2. Inst. of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland

    Peter Widmayer

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Bshouty, N.H. (2015). Linear Time Constructions of Some \(d\)-Restriction Problems. In: Paschos, V., Widmayer, P. (eds) Algorithms and Complexity. CIAC 2015. Lecture Notes in Computer Science(), vol 9079. Springer, Cham. https://doi.org/10.1007/978-3-319-18173-8_5

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  • DOI: https://doi.org/10.1007/978-3-319-18173-8_5

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