Advertisement

Optimal Monotone Encodings

  • Noga Alon
  • Rani Hod
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)

Abstract

Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2, ..., n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks.

To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multi-user tracing ((k, α)-FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n,k)-monotone encoding of length O(k log(n/k)).

We also present an explicit construction of an (n, 2)-monotone encoding of length 2logn + O(1), which is optimal up to an additive constant.

Keywords

Time Slot Explicit Construction Additive Constant Broadcast Scheme Probabilistic Construction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N.: Explicit construction of exponential sized families of k-independent sets. Discrete Mathematics 58(2), 191–193 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Asodi, V.: Tracing a single user. European Journal of Combinatorics 27(8), 1227–1234 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Asodi, V.: Tracing many users with almost no rate penalty. IEEE Transactions on Information Theory 53(1), 437–439 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Capalbo, M., Reingold, O., Vadhan, S., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: Proceedings of the 34th Annual ACM STOC, pp. 659–668 (2002)Google Scholar
  5. 5.
    Coppersmith, D., Shearer, J.: New bounds for union-free families of sets. Electronic Journal of Combinatorics 5(1), 39 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Csűrös, M., Ruszinkó, M.: Single user tracing and disjointly superimposed codes. IEEE Transactions on Information Theory 51(4), 1606–1611 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dyachkov, A.G., Rykov, V.V.: Bounds on the length of disjunctive codes. Problemy Peredachi Informatsii 18(3), 158–166 (1982)MathSciNetGoogle Scholar
  8. 8.
    Erdős, P., Frankl, P., Füredi, Z.: Families of finite sets in which no set is covered by the union of r others. Israel Journal of Mathematics 51(1-2), 79–89 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Füredi, Z.: A note on r-cover-free families. Journal of Combinatorial Theory Series A 73(1), 172–173 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guruswami, V., Umans, C., Vadhan, S.: Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. In: Proceedings of the 22nd IEEE CCC, pp. 96–108 (2007)Google Scholar
  11. 11.
    Komlós, J., Greenberg, A.: An asymptotically fast nonadaptive algorithm for conflict resolution in multiple-access channels. IEEE Transactions on Information Theory 31(2), 302–306 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Laczay, B., Ruszinkó, M.: Multiple user tracing codes. In: Proceedings of IEEE ISIT 2006, pp. 1900–1904 (2006)Google Scholar
  13. 13.
    Masser, D.W.: Note on a conjecture of Szpiro. Astérisque 183, 19–23 (1990)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Moran, T., Naor, M., Segev, G.: Deterministic History-Independent Strategies for Storing Information on Write-Once Memories. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 303–315. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Oesterlé, J.: Nouvelles approches du “théorème” de Fermat. Astérisque 161/162, 165–186 (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ruszinkó, M.: On the upper bound of the size of the r-cover-free families. Journal of Combinatorial Theory Series A 66(2), 302–310 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Noga Alon
    • 1
  • Rani Hod
    • 1
  1. 1.Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations