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Pseudo-Random Graphs and Bit Probe Schemes with One-Sided Error

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Abstract

We study probabilistic bit-probe schemes for the membership problem. Given a set A of at most n elements from the universe of size m we organize such a structure that queries of type “x∈A? ” can be answered very quickly. H. Buhrman, P.B. Miltersen, J. Radhakrishnan, and S. Venkatesh proposed a randomized bit-probe scheme that needs space of O(nlogm) bits. That scheme has a randomized algorithm processing queries; it needs to read only one randomly chosen bit from the memory to answer a query. For every x the answer is correct with high probability (with two-sided errors).

In this paper we slightly modify the bit-probe model of Buhrman et al. and consider schemes with a small auxiliary information in “cache” memory. In this model, we show that for the membership problem there exists a bit-probe scheme with one-sided error that needs space of O(nlog2 m+poly(logm)) bits, which cannot be achieved in the model without cache. We also obtain a slightly weaker result (space of size n 1+δpoly(logm) bits and two bit probes for every query) for a scheme that is effectively encodable.

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Notes

  1. The same argument can be presented in a more standard framework, with a read-once input tape and an index tape of poly-logarithmic size. However, we believe that the argument becomes more intuitive when we allow many passes on the input tape.

References

  1. Bloom, B.: Space-time trade-offs in hash coding with allowable errors. Commun. ACM 13(7), 422–426 (1970)

    Article  MATH  Google Scholar 

  2. Pinsker, M.S.: On the complexity of a concentrator. In: 7th International Teletrafc Conference, pp. 318/1–318/4 (1973)

    Google Scholar 

  3. Bassalygo, L.A., Pinsker, M.S.: The complexity of an optimal non-blocking commutation scheme without reorganization. Probl. Inf. Transm. 9, 64–66 (1974)

    Google Scholar 

  4. Dyachkov, A.G., Rykov, V.V.: Bounds on the length of disjunctive codes. Probl. Inf. Transm. 18(3), 7–13 (1982)

    MathSciNet  Google Scholar 

  5. Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach. 31(3), 538–544 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Erdös, P., Frankl, P., Füredi, Z.: Families of nite sets in which no set is covered by the union of r others. Isr. J. Math. 51, 79–89 (1985)

    Article  MATH  Google Scholar 

  7. Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Proc. of the 18th Annual ACM Symposium on Theory of Computing (STOC), pp. 6–20 (1986)

    Google Scholar 

  8. Siegel, A.: On universal classes of fast high performance hash functions, their time-space trade-off, and their applications. In: Proc. of 30th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 20–25 (1989)

    Chapter  Google Scholar 

  9. Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992). Preliminary version (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fiat, A., Naor, M., Schmidt, J.P., Siegel, A.: Non-oblivious hashing. J. ACM 39, 764–782 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nisan, N., Wigderson, A.: Hardness vs randomness. J. Comput. Syst. Sci. 49(2), 149–167 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kahale, N.: Eigenvalues and expansion of regular graphs. J. ACM 42(5), 1091–1106 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Siegel, A.: On universal classes of extremely random constant time hash functions and their time-space tradeoff. Technical report TR1995-684, Courant Institute, New York University, April (1995)

  14. Trevisan, L.: Construction of extractors using pseudo-random generators. In: Proc. of the 31st Annual ACM Symposium on Theory of Computing (STOC), pp. 141–148 (1999)

    Google Scholar 

  15. Brodnik, A., Munro, J.I.: Membership in constant time and minimum space. SIAM J. Comput. 28(5), 1627–1640 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pagh, R.: Low redundancy in static dictionaries with O(1) worst case lookup time. In: Proc. of the 26th International Colloquium on Automata, Languages and Programming (ICALP), pp. 595–604 (1999)

    Chapter  Google Scholar 

  17. Vitter, J.S.: External memory algorithms and data structures. ACM Comput. Surv. 33(2), 209–271 (2001)

    Article  Google Scholar 

  18. Buhrman, H., Miltersen, P.B., Radhakrishnan, J., Srinivasan, V.: Are bitvectors optimal? SIAM J. Comput. 31(6), 1723–1744 (2002). Preliminary version, pp. 449–458 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ta-Shma, A., Umans, C., Zuckerman, D.: Loss-less condensers, unbalanced expanders, and extractors. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 143–152 (2001)

    Google Scholar 

  20. Pagh, R.: On the cell probe complexity of membership and perfect hashing. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 425–432 (2001)

    Google Scholar 

  21. Capalbo, M.R., Reingold, O., Vadhan, S.P., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: Proc. of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp. 659–668 (2002)

    Google Scholar 

  22. Sivakumar, D.: Algorithmic derandomization via complexity theory. In: Proc. of the 34th Annual CM Symposium on Theory of Computing (STOC), pp. 619–626 (2002)

    Google Scholar 

  23. Östlin, A., Pagh, R.: One-Probe search. In: Proc. of the 29th International Colloquium on Automata, Languages and Programming (ICALP), pp. 439–450 (2002)

    Chapter  Google Scholar 

  24. Raz, R., Reingold, O., Vadhan, S.P.: Extracting all the randomness and reducing the error in Trevisan’s extractors. J. Comput. Syst. Sci. 65(1), 97–128 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ta-Shma, A.: Storing information with extractors. Inf. Process. Lett. 83, 267–274 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Devroye, L., Morin, P.: Cuckoo hashing: further analysis. Inf. Process. Lett. 86, 215–219 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pagh, R., Rodler, F.: Cuckoo hashing. J. Algorithms 51, 122–144 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  29. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Vitter, J.S.: Algorithms and Data Structures for External Memory. Series on Foundations and Trends in Theoretical Computer Science. Now Publishers, Hanover (2008)

    Google Scholar 

  31. Braverman, M.: Poly-logarithmic independence fools AC0 circuits. In: Proc. of the 27th IEEE Conference on Computational Complexity, pp. 3–8 (2009)

    Google Scholar 

  32. Arbitman, Y., Naor, M., Segev, G.: De-amortized cuckoo hashing: provable Worst-Case performance and experimental results. In: Proc. of the 36th International Colloquium on Automata, Languages and Programming. ICALP, vol. 1, pp. 107–118 (2009)

    Chapter  Google Scholar 

  33. Guruswami, V., Umans, C., Vadhan, S.: Unbalanced expanders and randomness extractors from Parvaresh–Vardy codes. J. ACM 56(4), 20:1–20:34 (2009)

    Article  MathSciNet  Google Scholar 

  34. Musatov, D.: Theorems about space-bounded Kolmogorov complexity obtained by “naive” derandomization. In: Proc. 6th International Computer Science Symposium (CSR), Russia, pp. 64–76 (2011)

    Google Scholar 

  35. David, M., Papakonstantinou, P.A., Sidiropoulos, A.: How strong is Nisan’s pseudorandom generator? Inf. Process. Lett. 111(6), 804–808 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

Supported in part by grants ANR EMC ANR-09-BLAN-0164-01 and NAFIT ANR-08-EMER-008-01.

The author thanks Daniil Musatov for useful discussions, and anonymous referees for deep and very helpful comments.

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  1. Univ. Montpellier 2, LIRMM UMR 5506-CC 477, 161 rue Ada, 34095, Montpellier, France

    Andrei Romashchenko

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  1. Andrei Romashchenko
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Correspondence to Andrei Romashchenko.

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A. Romashchenko on leave from IITP, Moscow.

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Romashchenko, A. Pseudo-Random Graphs and Bit Probe Schemes with One-Sided Error. Theory Comput Syst 55, 313–329 (2014). https://doi.org/10.1007/s00224-012-9425-0

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