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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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
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