Set membership with a few bit probes

We consider the bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form “Is x in S?” by adaptively probing the bit vector at t places. Let s(m,n, t) be the minimum number of bits of storage needed for such a scheme. Several recent works investigate s(m,n, t) for various ranges of the parameter; we obtain the following improvements over the bounds shown by Buhrman, Miltersen, Radhakrishnan, and Srinivasan [5] and Alon and Feige [2]. For two probes (t = 2): (a) s(m,n, 2) = O(m 1 4n+1 ); this improves on a result of Alon and Feige that states that for n ≤ lgm, s(m,n, 2) = O(mn lg((lgm)/n)/ lgm). (b) s(m,n, 2) = Ω(m 1 ⌊n/4⌋ ); in particular, s(m,n, 2) = Ω(m) for n ≥ lgm, that is, if s(m,n, 2) = o(m) (significantly better than the characteristic vector representation), then n = o(lgm). For three probes (t = 3): s(m,n, 3) = O( √ mn lg 2m n ). This improves a result of Alon and Feige that states that s(m,n, 2) = O(m 2 3n 1 3 ). In general: (a) (Non-adaptive schemes) For odd t ≥ 5, there is a non-adaptive scheme using O(tm 2 t−1 n 2 t−1 lg 2m n ) bits of space. This improves on a result of Buhrman et al. [5] that states that for odd t ≥ 5, there exists a non-adaptive scheme that uses O(tm 4 t+1n) bits of space. (b) (Adaptive schemes) For odd t ≥ 3 and t ≤ 1 10 lg lgm and for n ≤ m (ǫ > 0), we have s(m,n, t) = O(exp(e)m 2 t+1n 2 t+1 lgm). Previously, for t ≥ 5, no adaptive scheme was known that was more efficient than the non-adaptive scheme due to Buhrman et al. [5], which uses O(tm 4 t+1n) bits of space. (c) If t ≥ 3 and 4 ≤ n, then s(m,n, t) ≥ 1 15 m 1 t−1 (1− 4t n . For n ≤ lgm, this improves on the lower bound s(m,n, 3) = Ω( √ mn/ lgm) (valid only for n ≥ 16 lgm and for non-adaptive schemes) due to Alon and Feige; for small values of n, it also improves on the lower bound s(m,n, t) = Ω(tm 1 t n 1 t ) due to Buhrman et al. [5].