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.
Due to space limitations we refer the reader to a longer version available at http://www.math.tau.ac.il/~nogaa/PDFS/publications.html
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References
Alon, N.: Explicit construction of exponential sized families of k-independent sets. Discrete Mathematics 58(2), 191–193 (1986)
Alon, N., Asodi, V.: Tracing a single user. European Journal of Combinatorics 27(8), 1227–1234 (2006)
Alon, N., Asodi, V.: Tracing many users with almost no rate penalty. IEEE Transactions on Information Theory 53(1), 437–439 (2007)
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)
Coppersmith, D., Shearer, J.: New bounds for union-free families of sets. Electronic Journal of Combinatorics 5(1), 39 (1998)
Csűrös, M., Ruszinkó, M.: Single user tracing and disjointly superimposed codes. IEEE Transactions on Information Theory 51(4), 1606–1611 (2005)
Dyachkov, A.G., Rykov, V.V.: Bounds on the length of disjunctive codes. Problemy Peredachi Informatsii 18(3), 158–166 (1982)
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)
Füredi, Z.: A note on r-cover-free families. Journal of Combinatorial Theory Series A 73(1), 172–173 (1996)
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)
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)
Laczay, B., Ruszinkó, M.: Multiple user tracing codes. In: Proceedings of IEEE ISIT 2006, pp. 1900–1904 (2006)
Masser, D.W.: Note on a conjecture of Szpiro. Astérisque 183, 19–23 (1990)
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)
Oesterlé, J.: Nouvelles approches du “théorème” de Fermat. Astérisque 161/162, 165–186 (1988)
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)
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Alon, N., Hod, R. (2008). Optimal Monotone Encodings. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_22
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DOI: https://doi.org/10.1007/978-3-540-70575-8_22
Publisher Name: Springer, Berlin, Heidelberg
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