Optimal Monotone Encodings

  • Noga Alon
  • Rani Hod
Conference paper
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.

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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

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