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 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 alpha-fraction k -multiuser tracing ((k, alpha)-FUT (fraction user-tracing) families). We show that r(n, k) = Theta(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, alpha)-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 2 log n+O(1), which is optimal up to an additive constant.