Computer Science/Discrete Mathematics Seminar II

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-multi-user tracing ((k, alpha)-FUT 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. Joint work with Noga Alon.