Optimal Monotone Encodings

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 tokof {1, 2, …, n} to rbits.Monotone encodings are relevant to the study of tamper-proof datastructures and arise also in the design of broadcast schemes incertain communication networks. To answer this question, we develop a relaxation ofk-superimposed families, which we callα-fraction k-multi-user tracing((k, α)-FUT families). We show thatr(n, k) = θ(klog(n/k)) by proving tight asymptotic lower andupper bounds on the size of (k, α)-FUTfamilies and by constructing an (n,k)-monotoneencoding of length O(klog(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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